$g_{n,l,m}$ converges locally uniformly to some function $H_{l,m}$ harmonic in $\Omega_m$ as $\tilde{\mathcal{N}}_{m,l} \ni n \to \infty$ (note that $g_{n,l,m}$ cannot go to $+\infty$ otherwise so would $g(v_n, \Omega_l; \cdot)$, and in view of (3.34) $u_l(z)$ would be infinite for quasi every $z$, contradicting that $u_l \le u'' < \infty$). Of necessity, $H_{l,m} = u_l - u_m$ by (3.34), and a diagonal argument gives us a single subsequence $\tilde{\mathcal{N}}_m^* \subseteq \tilde{\mathcal{N}}$ along which the convergence $g_{n,l,m} \to H_{l,m}$ takes place for any $l > 0$. Now, for fixed $m$ and each $l > m$, select $\tilde{n}_l \in \tilde{\mathcal{N}}_m^*$ such that

$$ (3.40) \qquad |g_{n,l,m}(z) - (u_l(z) - u_m(z))| \le 1/l, \quad z \in \bar{\Omega}_{m-1}, \quad \tilde{n}_l \le n \in \tilde{\mathcal{N}}_m^*. $$

Since the functions $u_l - u_m$ are harmonic in $\Omega_m$ and increase with $l$, they converge locally uniformly to $u'' - u_m$ there by Harnack's theorem and the definition of $u''$. Thus, taking (3.40) into account, for any $\epsilon > 0$ there exists $L > 0$ such that

$$ (3.41) \qquad |g_{n,l,m}(z) - (u''(z) - u_m(z))| \le \epsilon, \quad z \in \bar{\Omega}_{m-1}, \quad l \ge L, \quad \tilde{n}_l \le n \in \tilde{\mathcal{N}}_m^*. $$

Define $\tilde{\mathcal{N}}_m := \{\tilde{n}_l\}_{l=1}^\infty$. Then it follows from (3.41) and (3.34) that

$$ (3.42) \qquad \liminf_{l \to \infty} g(v_{n_l}, \Omega_l; z) = \liminf_{l \to \infty} g(v_{n_l}, \Omega_m; z) + \lim_{l \to \infty} g_{n_l, l, m}(z) = u''(z) $$

for quasi every $z \in \bar{\Omega}_{m-1}$, whenever $\{n_l\}_{l=1}^\infty \subseteq \tilde{\mathcal{N}}_m$. Finally, let $\mathcal{N}''$ be the diagonal sequence of the table $\{\tilde{\mathcal{N}}_m\}_{m=1}^\infty$. Since $\mathcal{N}''$ is eventually a subsequence of every $\tilde{\mathcal{N}}_m$ it follows from (3.42) that

$$ (3.43) \qquad \liminf_{m \to \infty} g(v_{n_m}, \Omega_m; z) = u''(z), \quad \{n_m\}_{m=1}^\infty \subseteq \tilde{\mathcal{N}}'', \quad \text{for q.e. } z \in \mathcal{R} \setminus E_f, $$

which is the equality case in (3.29). $\square$

### 3.7. Logarithmic Error Function.
Hereafter, we redefine $\mathcal{N}$ to be $\mathcal{N}''$ constructed in Lemma 3.5. By Lemmas 3.1, 3.3 and 3.5, the limits (3.7), (3.12), (3.28), and (3.29) hold along this new sequence.

Since $\mu$ is finite, there is a $G_\delta$ polar set $\tilde{\mathcal{N}}_0 \subset \mathcal{D}$ such that $g(\mu, \mathcal{D}; x) < +\infty$ for $x \in \mathcal{D} \setminus \tilde{\mathcal{N}}_0$, see Sections A.3 and A.5. Let us put $N_0 := p^{-1}(\tilde{\mathcal{N}}_0)$, which is a $G_\delta$ polar subset of $\mathcal{R}$, see Section A.5. We now introduce the function $\text{ler} : \mathcal{R}\setminus N_0 \to [-\infty, +\infty)$ ("ler" for "logarithmic error"), by putting

$$ (3.44) \qquad \text{ler}(z) := g(\mu, \mathcal{D}; p(z)) - u'(z) - u''(z), \quad z \in \mathcal{R}\setminus N_0, $$

where $u'$ and $u''$ are as in Lemmas 3.1 and 3.5. Clearly, $\text{ler}(z)$ is a $\delta$-subharmonic function (the difference of two subharmonic functions), and it is well defined for $z \notin N_0$ since $g(\mu, \mathcal{D}; p(z))$ is finite there. As introduced, $\text{ler}$ depends on the choice of the subsequence $\mathcal{N}$, but later we shall see that it is in fact unique.

**Lemma 3.6.** There exists a polar set $A_0 \subset \mathcal{R}\setminus N_0$ such that, whenever $z_1, z_2$ are distinct points in $\mathcal{R}\setminus N_0$ with $p(z_1) = p(z_2)$ and $\text{ler}(z_i) < 0$ for $i = 1, 2$, then $z_1, z_2 \in A_0$.

*Proof.* By the equality quasi everywhere in (3.28), there exists a polar set $A_1 \subset \mathcal{D}$ such that, for every $x \in \mathcal{D} \setminus A_1$, one can find a sequence $\mathcal{N}_x \subseteq \mathcal{N}$ along which

$$ \lim_{\mathcal{N}_x \ni n \to \infty} \frac{1}{n} \log |b_n^{\text{pole}}(x)|^{-1} = g(\mu, \mathcal{D}; x). $$

Together with (3.11), (3.12), the inequality in (3.29) and (3.44), this gives us

$$ (3.45) \qquad \limsup_{\mathcal{N}_x \ni n \to \infty} \frac{1}{n} \log |(f - N(M_n) \circ p)(z)| \le \text{ler}(z) $$

for every $z \notin N_0$ such that $z \in p^{-1}(x)$ with $x \notin A_1 \cup p(E_f)$. Assume now that $z_1, z_2 \in \mathcal{R}\setminus N_0$ satisfy $z_1 \ne z_2$ and $p(z_1) = p(z_2)$, as well as $\text{ler}(z_i) < 0$ for $i = 1, 2$. Let $x \in \mathcal{D}$ be such that $z_1, z_2 \in p^{-1}(x)$. If $x \notin A_1 \cup p(E_f)$, it follows from (3.45) that

$$ f(z_1) = \lim_{\mathcal{N}_x \ni n \to \infty} N(M_n)(p(z_1)) = \lim_{\mathcal{N}_x \ni n \to \infty} N(M_n)(p(z_2)) = f(z_2) $$
and necessarily $z_1, z_2 \in A_2 := \{\zeta | \exists \eta \neq \zeta : p(\zeta) = p(\eta), f(\zeta) = f(\eta)\}$. Now, the conditions placed on the class $\mathcal{F}(\mathcal{R})$ imply that the set $A_2$ is finite, and therefore the lemma holds with $A_0 := p^{-1}(A_1 \cup p(E_f) \cup p(A_2))$ which is polar, as inverse image under $p$ of a polar set. $\square$

**Lemma 3.7.** *The inequality $ler(z) > -\infty$ holds for quasi every $z \in \mathcal{R}\backslash N_0$. In particular $u' \neq +\infty$ and $u'' \neq +\infty$ in Lemmas 3.1 and 3.5, so $h'$ and $h''$ are finite non-negative harmonic functions on $\mathcal{R}$.*

*Proof.* Since $g(\mu; \mathcal{D}; p(\cdot))$ is finite on $\mathcal{R}\backslash N_0$ while $u', u''$ are non-negative and either identically $+\infty$ or finite quasi everywhere, $ler$ is in turn either identically $-\infty$ or finite quasi everywhere on $\mathcal{R}\backslash N_0$. The former possibility contradicts Lemma 3.6, therefore the latter prevails so that $u' \neq +\infty$ and $u'' \neq +\infty$. Hence, Lemmas 3.1 and 3.5 imply that $h'$ and $h''$ are finite on $\mathcal{R}$. $\square$

Due to the previous lemma, we can rewrite (3.44) as

$$ (3.46) \qquad ler(z) = g(\mu, \mathcal{D}; p(z)) - g(v, \mathcal{R}; z) - h_{\mathcal{R}}(z), \quad z \in \mathcal{R}\backslash N_0, $$

where we have set $v := v' + v''$, which is a locally finite measure on $\mathcal{R}$ with quasi everywhere finite potential, and $h_{\mathcal{R}} := h' + h''$ which is a positive harmonic function on $\mathcal{R}$.

**Lemma 3.8.** *It holds that $\lim_{z \to \zeta} h_{\mathcal{R}}(z) = 0$ for every $\zeta \in \partial \mathcal{R} \setminus \mathcal{T}$.*

*Proof.* Fix $\zeta \in \partial \mathcal{R} \setminus \mathcal{T}$, and let $\xi \in \mathcal{T}$ be such that $p(\zeta) = p(\xi)$. We claim that

$$ (3.47) \qquad \liminf_{z \to \zeta} h_{\mathcal{R}}(z) = 0. $$

Indeed, if $\liminf_{z \to \zeta} h_{\mathcal{R}}(z) = l > 0$, take $0 < 2\epsilon := \min\{l, 2/\text{cap}_{\mathcal{D}}(K_f)\}$. Let $S \subset \mathcal{D}$ be the radial segment $\{z : z = rp(\zeta), r \in [1 - \delta, 1)\}$ and $S_{\zeta}$ (resp. $S_{\xi}$) be the connected component of $p^{-1}(S)$ accumulating on $\partial \mathcal{R}$ to $\zeta$ (resp. $\xi$). If $\delta > 0$ is small enough, then

$$ (3.48) \qquad h_{\mathcal{R}}(z) \geq \epsilon, \quad z \in S_{\zeta} \cup S_{\xi}, $$

by Lemma 3.2. Furthermore, Lemma 3.6 yields that either $ler(z_1) \geq 0$ or $ler(z_2) \geq 0$ if $p(z_1) = p(z_2) \in S \setminus p(A_0 \cup N_0)$. In particular, we get from (3.46) and (3.48) that

$$ (3.49) \qquad g(\mu, \mathcal{D}; z) \geq \epsilon, \quad z \in S \setminus p(A_0 \cup N_0), $$

where we notice that $A_0 \cup N_0$ as well as $p(A_0 \cup N_0)$ are polar. This contradicts Lemma A.5, applied with $g(\sigma, D; \cdot) = g(\mu, D; \cdot)$ and $\xi$ being $p(\zeta)$, since $R_{\epsilon/2}$ from that lemma would necessarily be a subset of $p(A_0 \cup N_0)$. This proves our claim (3.47).

Next, assume for a contradiction that $\limsup_{z \to \zeta} h_{\mathcal{R}}(z) = l' > 0$, and pick $0 < 2\epsilon \le \min\{l', 2/\text{cap}_{\mathcal{D}}(K_f)\}$ such that the level line $L_{\epsilon} := \{z : h_{\mathcal{R}}(z) = \epsilon\}$ is a smooth 1-dimensional submanifold of $\mathcal{R}$ (this can be achieved according to Sard's theorem). Notice that $\zeta$ must be a limit point of $L_{\epsilon}$ because any neighborhood of $\zeta$ in $\mathcal{R}_*$ contains a connected open set $U \ni \zeta$ with $U \cap \mathcal{R}$ connected ($p$ is a local homeomorphism at $\zeta$ and we may take $p(U)$ to be a disk) in which $h_{\mathcal{R}}$ assumes values arbitrary close to 0 and $l'$ by definition of $\liminf$ and $\limsup$; hence, as $h_{\mathcal{R}}(U \cap \mathcal{R})$ is connected, it contains the value $\epsilon$.

Let $D_0$ be a disk centered at $p(\zeta)$ of small enough radius so that $D_{\zeta}$ and $D_{\xi}$, the components of $p^{-1}(D_0)$ in $\mathcal{R}_*$ that contain respectively $\zeta$ and $\xi$, are in one-to-one correspondence with $D_0$ under $p$. Decreasing the radius of $D_0$ if necessary, we can assume that $h_{\mathcal{R}}(z) \ge \epsilon/2$ for $z \in D_{\xi}$ by Lemma 3.2. Let us redefine

$$ (3.50) \qquad S_{\zeta} := L_{\epsilon} \cap \bar{D}_{\zeta}, \quad S := p(S_{\zeta}), \quad \text{and} \quad S_{\xi} := p^{-1}(S) \cap \bar{D}_{\xi}. $$

Observe that $h_{\mathcal{R}}$ cannot be constant in view of (3.47) and (3.17). Therefore, no connected component of $S$ can be a closed curve in $D_0$ by the maximum principle for harmonic functions. In addition, if $S$ has a connected component, say $S_*$, accumulating at $z_* \in \mathcal{T}$, then

$$ g(\mu, \mathcal{D}; z) \geq \epsilon/2, \quad z \in S_* \setminus p(A_0 \cup N_0), $$
and necessarily $z_1, z_2 \in A_2 := \{\zeta | \exists \eta \neq \zeta : p(\zeta) = p(\eta), f(\zeta) = f(\eta)\}$. Now, the conditions placed on the class $\mathcal{F}(\mathcal{R})$ imply that the set $A_2$ is finite, and therefore the lemma holds with $A_0 := p^{-1}(A_1 \cup p(E_f) \cup p(A_2))$ which is polar, as inverse image under $p$ of a polar set. $\square$

**Lemma 3.7.** *The inequality $ler(z) > -\infty$ holds for quasi every $z \in \mathcal{R}\backslash N_0$. In particular $u' \neq +\infty$ and $u'' \neq +\infty$ in Lemmas 3.1 and 3.5, so $h'$ and $h''$ are finite non-negative harmonic functions on $\mathcal{R}$.*

*Proof.* Since $g(\mu; \mathcal{D}; p(\cdot))$ is finite on $\mathcal{R}\backslash N_0$ while $u', u''$ are non-negative and either identically $+\infty$ or finite quasi everywhere, $ler$ is in turn either identically $-\infty$ or finite quasi everywhere on $\mathcal{R}\backslash N_0$. The former possibility contradicts Lemma 3.6, therefore the latter prevails so that $u' \neq +\infty$ and $u'' \neq +\infty$. Hence, Lemmas 3.1 and 3.5 imply that $h'$ and $h''$ are finite on $\mathcal{R}$. $\square$

Due to the previous lemma, we can rewrite (3.44) as

$$ (3.46) \qquad ler(z) = g(\mu, \mathcal{D}; p(z)) - g(v, \mathcal{R}; z) - h_{\mathcal{R}}(z), \quad z \in \mathcal{R}\backslash N_0, $$

where we have set $v := v' + v''$, which is a locally finite measure on $\mathcal{R}$ with quasi everywhere finite potential, and $h_{\mathcal{R}} := h' + h''$ which is a positive harmonic function on $\mathcal{R}$.

**Lemma 3.8.** *It holds that $\lim_{z \to \zeta} h_{\mathcal{R}}(z) = 0$ for every $\zeta \in \partial \mathcal{R} \setminus \mathcal{T}$.*

*Proof.* Fix $\zeta \in \partial \mathcal{R} \setminus \mathcal{T}$, and let $\xi \in \mathcal{T}$ be such that $p(\zeta) = p(\xi)$. We claim that

$$ (3.47) \qquad \liminf_{z \to \zeta} h_{\mathcal{R}}(z) = 0. $$

Indeed, if $\liminf_{z \to \zeta} h_{\mathcal{R}}(z) = l > 0$, take $0 < 2\epsilon := \min\{l, 2/\text{cap}_{\mathcal{D}}(K_f)\}$. Let $S \subset \mathcal{D}$ be the radial segment $\{z : z = rp(\zeta), r \in [1 - \delta, 1)\}$ and $S_{\zeta}$ (resp. $S_{\xi}$) be the connected component of $p^{-1}(S)$ accumulating on $\partial \mathcal{R}$ to $\zeta$ (resp. $\xi$). If $\delta > 0$ is small enough, then

$$ (3.48) \qquad h_{\mathcal{R}}(z) \geq \epsilon, \quad z \in S_{\zeta} \cup S_{\xi}, $$

by Lemma 3.2. Furthermore, Lemma 3.6 yields that either $ler(z_1) \geq 0$ or $ler(z_2) \geq 0$ if $p(z_1) = p(z_2) \in S \setminus p(A_0 \cup N_0)$. In particular, we get from (3.46) and (3.48) that

$$ (3.49) \qquad g(\mu, \mathcal{D}; z) \geq \epsilon, \quad z \in S \setminus p(A_0 \cup N_0), $$

where we notice that $A_0 \cup N_0$ as well as $p(A_0 \cup N_0)$ are polar. This contradicts Lemma A.5, applied with $g(\sigma, D; \cdot) = g(\mu, D; \cdot)$ and $\xi$ being $p(\zeta)$, since $R_{\epsilon/2}$ from that lemma would necessarily be a subset of $p(A_0 \cup N_0)$. This proves our claim (3.47).

Next, assume for a contradiction that $\limsup_{z \to \zeta} h_{\mathcal{R}}(z) = l' > 0$, and pick $0 < 2\epsilon \le \min\{l', 2/\text{cap}_{\mathcal{D}}(K_f)\}$ such that the level line $L_{\epsilon} := \{z : h_{\mathcal{R}}(z) = \epsilon\}$ is a smooth 1-dimensional submanifold of $\mathcal{R}$ (this can be achieved according to Sard's theorem). Notice that $\zeta$ must be a limit point of $L_{\epsilon}$ because any neighborhood of $\zeta$ in $\mathcal{R}_*$ contains a connected open set $U \ni \zeta$ with $U \cap \mathcal{R}$ connected ($p$ is a local homeomorphism at $\zeta$ and we may take $p(U)$ to be a disk) in which $h_{\mathcal{R}}$ assumes values arbitrary close to 0 and $l'$ by definition of $\liminf$ and $\limsup$; hence, as $h_{\mathcal{R}}(U \cap \mathcal{R})$ is connected, it contains the value $\epsilon$.

Let $D_0$ be a disk centered at $p(\zeta)$ of small enough radius so that $D_{\zeta}$ and $D_{\xi}$, the components of $p^{-1}(D_0)$ in $\mathcal{R}_*$ that contain respectively $\zeta$ and $\xi$, are in one-to-one correspondence with $D_0$ under $p$. Decreasing the radius of $D_0$ if necessary, we can assume that $h_{\mathcal{R}}(z) \ge \epsilon/2$ for $z \in D_{\xi}$ by Lemma 3.2. Let us redefine

$$ (3.50) \qquad S_{\zeta} := L_{\epsilon} \cap \bar{D}_{\zeta}, \quad S := p(S_{\zeta}), \quad \text{and} \quad S_{\xi} := p^{-1}(S) \cap \bar{D}_{\xi}. $$

Observe that $h_{\mathcal{R}}$ cannot be constant in view of (3.47) and (3.17). Therefore, no connected component of $S$ can be a closed curve in $D_0$ by the maximum principle for harmonic functions. In addition, if $S$ has a connected component, say $S_*$, accumulating at $z_* \in \mathcal{T}$, then

$$ g(\mu, \mathcal{D}; z) \geq \epsilon/2, \quad z \in S_* \setminus p(A_0 \cup N_0), $$
exactly as in (3.49). Let $R_{\epsilon/3}$ be as in Lemma A.5, applied with $g(\sigma, D; \cdot) = g(\mu, D; \cdot)$ and $\xi$ being $z_*$. Then $S_*$ must intersect every circle $\{z \in D : |z - z_*| = 1 - r\}, r \in R_{\epsilon/3}$, by connectedness. Necessarily, the intersection must be a subset of $p(A_0 \cup N_0)$ and therefore polar. Since contractive maps do not increase the logarithmic capacity [53, Theorem 5.3.1], this means that $R_{\epsilon/3}$ is polar, which contradicts Lemma A.5 (as we explain in Section A.4, polar subsets of $D$ have Greenian and logarithmic outer capacity zero). Thus, $S$ is a system of smooth curves, each connected component of which has at least one limit point on $\partial D_0 \cap D$. Consequently, if $T_0 \subset D_0$ is a circle centered at $p(\zeta)$, then any connected component of $S$ intersecting the interior of $T_0$ must intersect $T_0$ as well since it accumulates at a point of $\partial D_0$. That is, $S$ must intersect any such circle and we arrive at a contradiction exactly as above. $\square$

The exceptional set where inequality is strict in Lemmas 3.3 and 3.5 a priori depends on the
subsequence $\{n_m\}$ under consideration. The next lemma shows that there exists a polar set outside
of which equality holds, both in (3.28) and (3.29), for one and the same subsequence, see (3.51).

**Lemma 3.9.** For quasi every $z \in \mathbb{R}$, there is a sequence $N_z = \{n_m^z\}_{m=1}^\infty \subseteq \mathcal{N}$ such that

$$
\left\{
\begin{array}{l}
\displaystyle \lim_{m \to \infty} g(v_{n_m^z}, \Omega_m; z) = u''(z), \\
\\
\displaystyle \lim_{m \to \infty} g(\mu_{n_m^z}, \mathbb{D}; p(z)) = g(\mu, \mathbb{D}; p(z)).
\end{array}
\right.
$$

*Proof.* Our goal is to show that there exists a subsequence $\{n_m^*\} \subseteq \mathcal{N}$ such that

$$
(3.51) \quad \liminf_{m \to \infty} \left( g(v_{n_m^*}, \Omega_m; z) + g(\mu_{n_m^*}, \mathbb{D}; p(z)) \right) = g(\mu, \mathbb{D}; p(z)) + u''(z)
$$

for quasi every $z \in \mathbb{R}$. Since the inequalities in (3.28) and (3.29) hold for every $z \in \mathbb{R}$, this will
indeed imply the claim of the lemma.

To prove (3.51), we shall rewrite the sum of two potentials in the left-hand side as a single potential on $\mathcal{R}$. To this end, we lift $\mu_n$ and $\mu$ to $\mathcal{R}$ via the construction described in (A.33). Specifically, with the notation introduced there, it follows from (A.34) that

$$
(3.52) \quad g(\hat{\mu}_n, \mathcal{R}; z) = g(\mu_n, \mathcal{D}; p(z)) \quad \text{and} \quad g(\hat{\mu}, \mathcal{R}; z) = g(\mu, \mathcal{D}; p(z)), \quad z \in \mathcal{R}.
$$

Now, we can write $g(v_n, \Omega_m; z) + g(\mu_n, \mathbb{D}; p(z))$ as a sum of three terms:

$$
(3.53) \quad g(v_n + \hat{\mu}_n, \Omega_m; z) + \left( g(\hat{\mu}_{n|\Omega_m}, \mathcal{R}, z) - g(\hat{\mu}_n, \Omega_m; z) \right) + g(\hat{\mu}_{n|\mathcal{R}\Omega_m}, \mathcal{R}; z),
$$

and we shall study their behavior separately.

To start, recall that $\mu = \mu'_{\mathbb{D}}$, where $\mu'$ is the weak* limit of $\mu_n$ along $\mathcal{N}$ in $\bar{\mathbb{D}}$. Thus, by the discussion after (A.34), an analogous relation holds between $\hat{\mu}$ and $\hat{\mu}_n$. Namely, since $\mu_n$ has total mass 1, the total mass of $\hat{\mu}_n$ is equal to $M$, the number of sheets of $\bar{\mathcal{R}}$. In particular, the sequence $\hat{\mu}_n$ converges weak* on $\bar{\mathcal{R}}$ to $\hat{\mu}'$, and on $\mathcal{R}$ to $\hat{\mu}'_{\mathcal{R}} = \hat{\mu}$.

Since the sets $\Omega_m$ exhaust $\mathcal{R}\E_f$, it holds that $\hat{\mu}(\mathcal{R}\\Omega_m) - \hat{\mu}(E_f) \to 0$ as $m \to \infty$. Moreover, as each $\mathcal{R}\\Omega_m$ is compact with boundary of $\hat{\mu}$-measure zero, it follows from Lemma 3.4 that the measures $\hat{\mu}_{n|\mathcal{R}\\Omega_m}$ converge weak* to $\hat{\mu}_{\mathcal{R}\\Omega_m}$ along $\mathcal{N}$. In particular, $\hat{\mu}_n(\mathcal{R}\\Omega_m) - \hat{\mu}(\mathcal{R}\\Omega_m) \to 0$ as $n \to \infty$. Hence, for each $m$ there exists $n'_m \in \mathcal{N}$ such that

$$
|\hat{\mu}_n(\mathcal{R}\setminus\Omega_m) - \hat{\mu}(E_f)| \le 2|\hat{\mu}(\mathcal{R}\setminus\Omega_m) - \hat{\mu}(E_f)| \quad \text{as soon as} \quad n \ge n'_m.
$$

Let $\sigma$ be a weak* limit point on $\mathcal{R}$ of the family $\{\hat{\mu}_{n'_m|\mathcal{R}\setminus\Omega_m}\}_{m\in\mathbb{N}}$. Clearly $\text{supp}(\sigma) \subseteq E_f$ and $\sigma$ is also a weak* limit point of this family on the fixed compact set $\mathcal{R}\setminus\Omega_1$. Thus, integrating against any function which is identically 1 on $\mathcal{R}\setminus\Omega_1$ and passing to the limit gives us $\sigma(E_f) = \hat{\mu}(E_f)$ by our very choice of $n'_m$. In another connection, if $\varphi: \mathcal{R} \to [0, 1]$ is a continuous function with compact support which is 1 on a compact set $K \subseteq E_f$, then

$$
\sigma(K) \le \int \varphi d\sigma \le \limsup_m \int \varphi d\hat{\mu}_{n'_m|\mathcal{R}\setminus\Omega_m} \le \lim_m \int \varphi d\hat{\mu}_{n'_m} = \int \varphi d\hat{\mu}.
$$
exactly as in (3.49). Let $R_{\epsilon/3}$ be as in Lemma A.5, applied with $g(\sigma, D; \cdot) = g(\mu, D; \cdot)$ and $\xi$ being $z_*$. Then $S_*$ must intersect every circle $\{z \in D : |z - z_*| = 1 - r\}, r \in R_{\epsilon/3}$, by connectedness. Necessarily, the intersection must be a subset of $p(A_0 \cup N_0)$ and therefore polar. Since contractive maps do not increase the logarithmic capacity [53, Theorem 5.3.1], this means that $R_{\epsilon/3}$ is polar, which contradicts Lemma A.5 (as we explain in Section A.4, polar subsets of $D$ have Greenian and logarithmic outer capacity zero). Thus, $S$ is a system of smooth curves, each connected component of which has at least one limit point on $\partial D_0 \cap D$. Consequently, if $T_0 \subset D_0$ is a circle centered at $p(\zeta)$, then any connected component of $S$ intersecting the interior of $T_0$ must intersect $T_0$ as well since it accumulates at a point of $\partial D_0$. That is, $S$ must intersect any such circle and we arrive at a contradiction exactly as above. $\square$

The exceptional set where inequality is strict in Lemmas 3.3 and 3.5 a priori depends on the
subsequence $\{n_m\}$ under consideration. The next lemma shows that there exists a polar set outside
of which equality holds, both in (3.28) and (3.29), for one and the same subsequence, see (3.51).

**Lemma 3.9.** For quasi every $z \in \mathbb{R}$, there is a sequence $N_z = \{n_m^z\}_{m=1}^\infty \subseteq \mathcal{N}$ such that

$$
\left\{
\begin{array}{l}
\displaystyle \lim_{m \to \infty} g(v_{n_m^z}, \Omega_m; z) = u''(z), \\
\\
\displaystyle \lim_{m \to \infty} g(\mu_{n_m^z}, \mathbb{D}; p(z)) = g(\mu, \mathbb{D}; p(z)).
\end{array}
\right.
$$

*Proof.* Our goal is to show that there exists a subsequence $\{n_m^*\} \subseteq \mathcal{N}$ such that

$$
(3.51) \quad \liminf_{m \to \infty} \left( g(v_{n_m^*}, \Omega_m; z) + g(\mu_{n_m^*}, \mathbb{D}; p(z)) \right) = g(\mu, \mathbb{D}; p(z)) + u''(z)
$$

for quasi every $z \in \mathbb{R}$. Since the inequalities in (3.28) and (3.29) hold for every $z \in \mathbb{R}$, this will
indeed imply the claim of the lemma.

To prove (3.51), we shall rewrite the sum of two potentials in the left-hand side as a single potential on $\mathcal{R}$. To this end, we lift $\mu_n$ and $\mu$ to $\mathcal{R}$ via the construction described in (A.33). Specifically, with the notation introduced there, it follows from (A.34) that

$$
(3.52) \quad g(\hat{\mu}_n, \mathcal{R}; z) = g(\mu_n, \mathcal{D}; p(z)) \quad \text{and} \quad g(\hat{\mu}, \mathcal{R}; z) = g(\mu, \mathcal{D}; p(z)), \quad z \in \mathcal{R}.
$$

Now, we can write $g(v_n, \Omega_m; z) + g(\mu_n, \mathbb{D}; p(z))$ as a sum of three terms:

$$
(3.53) \quad g(v_n + \hat{\mu}_n, \Omega_m; z) + \left( g(\hat{\mu}_{n|\Omega_m}, \mathcal{R}, z) - g(\hat{\mu}_n, \Omega_m; z) \right) + g(\hat{\mu}_{n|\mathcal{R}\Omega_m}, \mathcal{R}; z),
$$

and we shall study their behavior separately.

To start, recall that $\mu = \mu'_{\mathbb{D}}$, where $\mu'$ is the weak* limit of $\mu_n$ along $\mathcal{N}$ in $\bar{\mathbb{D}}$. Thus, by the discussion after (A.34), an analogous relation holds between $\hat{\mu}$ and $\hat{\mu}_n$. Namely, since $\mu_n$ has total mass 1, the total mass of $\hat{\mu}_n$ is equal to $M$, the number of sheets of $\bar{\mathcal{R}}$. In particular, the sequence $\hat{\mu}_n$ converges weak* on $\bar{\mathcal{R}}$ to $\hat{\mu}'$, and on $\mathcal{R}$ to $\hat{\mu}'_{\mathcal{R}} = \hat{\mu}$.

Since the sets $\Omega_m$ exhaust $\mathcal{R}\E_f$, it holds that $\hat{\mu}(\mathcal{R}\\Omega_m) - \hat{\mu}(E_f) \to 0$ as $m \to \infty$. Moreover, as each $\mathcal{R}\\Omega_m$ is compact with boundary of $\hat{\mu}$-measure zero, it follows from Lemma 3.4 that the measures $\hat{\mu}_{n|\mathcal{R}\\Omega_m}$ converge weak* to $\hat{\mu}_{\mathcal{R}\\Omega_m}$ along $\mathcal{N}$. In particular, $\hat{\mu}_n(\mathcal{R}\\Omega_m) - \hat{\mu}(\mathcal{R}\\Omega_m) \to 0$ as $n \to \infty$. Hence, for each $m$ there exists $n'_m \in \mathcal{N}$ such that

$$
|\hat{\mu}_n(\mathcal{R}\setminus\Omega_m) - \hat{\mu}(E_f)| \le 2|\hat{\mu}(\mathcal{R}\setminus\Omega_m) - \hat{\mu}(E_f)| \quad \text{as soon as} \quad n \ge n'_m.
$$

Let $\sigma$ be a weak* limit point on $\mathcal{R}$ of the family $\{\hat{\mu}_{n'_m|\mathcal{R}\setminus\Omega_m}\}_{m\in\mathbb{N}}$. Clearly $\text{supp}(\sigma) \subseteq E_f$ and $\sigma$ is also a weak* limit point of this family on the fixed compact set $\mathcal{R}\setminus\Omega_1$. Thus, integrating against any function which is identically 1 on $\mathcal{R}\setminus\Omega_1$ and passing to the limit gives us $\sigma(E_f) = \hat{\mu}(E_f)$ by our very choice of $n'_m$. In another connection, if $\varphi: \mathcal{R} \to [0, 1]$ is a continuous function with compact support which is 1 on a compact set $K \subseteq E_f$, then

$$
\sigma(K) \le \int \varphi d\sigma \le \limsup_m \int \varphi d\hat{\mu}_{n'_m|\mathcal{R}\setminus\Omega_m} \le \lim_m \int \varphi d\hat{\mu}_{n'_m} = \int \varphi d\hat{\mu}.
$$
As the infimum over $\varphi$ of the rightmost term above is $\hat{\mu}(K)$ by Urysohn's lemma and the outer regularity of Radon measures, we get that $\sigma \le \hat{\mu}_{\lfloor E_f}$ by the inner regularity of Radon measures on $\mathcal{R}$, see [54, Theorem 2.18]. Altogether, we deduce that $\sigma = \hat{\mu}_{\lfloor E_f}$, and consequently the measures $\hat{\mu}_{n'_m \lfloor \mathcal{R} \setminus \Omega_m}$ converge weak* to $\hat{\mu}_{\lfloor E_f}$ along $\mathcal{N}' := \{n'_m\}$. In particular,

$$ (3.54) \qquad \lim_{m \to \infty} g(\hat{\mu}_{n'_m \lfloor \mathcal{R} \setminus \Omega_m}, \mathcal{R}; z) = g(\hat{\mu}_{\lfloor E_f}, \mathcal{R}; z), \quad z \in \mathcal{R} \setminus E_f, $$

which settles the asymptotic behavior of the last term in (3.53) along the sequence of indices $\{n'_m\}_m$. Notice that by the definition of $n'_m$, if $\mathcal{N}'$ is replaced by an arbitrary subsequence $\{n''_m\}_m$ thereof, then (3.54) continues to hold along this new subsequence of indices. This will be used in the forthcoming steps.

Next, let $f_m(z, w) := g_{\mathcal{R}}(z, w) - g_{\Omega_m}(z, w)$ for $z, w \in \Omega_m$. Clearly, $f_m(z, \cdot)$ is harmonic in $\Omega_m$ and continuous on $\bar{\Omega}_m$, by the regularity of $\Omega_m$. Its boundary values are equal to $g_{\mathcal{R}}(z, \cdot)$ on $\partial\Omega_m$, in particular they are identically zero on $\partial\mathcal{R}$. Fix $z \in \mathcal{R} \setminus E_f$, and let $k$ be an integer such that $z \in \Omega_k$. Then, we get for all $m > k$ that

$$ (3.55) \qquad 0 \le f_m(z, w) \le \max_{w \in \partial \Omega_m \setminus \partial \mathcal{R}} g_{\mathcal{R}}(z, w) \le \max_{w \in \partial \Omega_{k+1}} g_{\mathcal{R}}(z, w) :=: C_k, \quad w \in \bar{\Omega}_m, $$

where the constant $C_k$ is finite, independent of $m$, and we used the maximum principle for harmonic functions twice (once for $f_m(z, \cdot)$ on $\Omega_m$ and once for $g_{\mathcal{R}}(z, \cdot)$ on $\mathcal{R} \setminus \Omega_{k+1}$). Observe further that the functions $f_m(z, \cdot)$, $m > k$, are not only positive harmonic in each $\Omega_l$ for fixed $l$ satisfying $k < l \le m$, but form a decreasing sequence there. Therefore they converge in $\Omega_l$ to a non-negative harmonic function, say $f^{\{l\}}(z, \cdot)$, by Harnack's theorem. As this claim is true for all large $l$, the $f^{\{l\}}(z, \cdot)$ inductively define a harmonic function $f(z, \cdot)$ on $\mathcal{R} \setminus E_f$ that satisfies $0 \le f(z, \cdot) \le C_k$ when $z \in \Omega_k$. Thus, it extends harmonically to the entire surface $\mathcal{R}$ by the Removability theorem for harmonic functions, and as its trace on $\partial\mathcal{R}$ is zero we conclude that $f(z, \cdot) = 0$.

Observe now that $f_m(z, w) = f_m(w, z)$ for $m$ large enough so that $z, w \in \Omega_m$. Thus, it is jointly harmonic in both variables [35, p. 561]. Hence, by Harnack's theorem and the diagonal argument, any subsequence of $\{f_m(\cdot, \cdot)\}$ has a further subsequence converging locally uniformly in $\mathcal{R} \setminus E_f \times \mathcal{R} \setminus E_f$, and we know from what precedes that the limit function can only be zero. In particular, it holds that

$$ (3.56) \qquad \eta_{l,m} := \max_{z,w \in \Omega_l} f_m(z,w) \to 0 \quad \text{as} \quad m \to \infty, $$

where we used that each $f_m(\cdot, \cdot)$ extends continuously by zero to $\partial\mathcal{R} \times \partial\mathcal{R}$, and therefore the maximum principle for harmonic functions can be applied to show that the convergence is indeed uniform on $\bar{\Omega}_l$. Given $m$, define

$$ l_m := \max \{l : 1 \le l \le m, \eta_{l,m} \le 1/l\}. $$

It follows from (3.56) that $l_m \to \infty$ and $\eta_{l_m, m} \to 0$ when $m \to \infty$, whence for each fixed $z \in \Omega_k$ and all $m > k$ it holds that

$$ (3.57) \qquad 0 \le \int_{\Omega_m} f_m(z, w) d\hat{\mu}_n(w) = \left( \int_{\Omega_m \setminus \Omega_{l_m}} + \int_{\Omega_{l_m}} \right) f_m(z, w) d\hat{\mu}_n(w) \\ \le C_k \hat{\mu}_n(\bar{\Omega}_m \setminus \Omega_{l_m}) + M \eta_{l_m, m}, $$

where we used (3.55) while recalling that $M$ is the number of sheets of $\mathcal{R}$, which is a bound for the total mass of each $\hat{\mu}_n$. Now, we obtain from Lemma 3.4 and our choice of $\Omega_m$ that the measures $\hat{\mu}_n$ converge weak* to $\hat{\mu}$ along $\mathcal{N}$, not only on $\mathcal{R}$, but also on every compact set of the form $\bar{\Omega}_m \setminus \Omega_{l_m}$. Hence, taking into account that $\hat{\mu}((\mathcal{R} \setminus E_f) \setminus \Omega_{l_m}) \to 0$ as $m \to \infty$, we can associate to each $m$ an integer $n''_m \in \mathcal{N}'$ such that $n''_m \ge n'_m$ and

$$ \hat{\mu}_{n''_m}(\bar{\Omega}_m \setminus \Omega_{l_m}) \to 0 \quad \text{as} \quad m \to \infty. $$
As the infimum over $\varphi$ of the rightmost term above is $\hat{\mu}(K)$ by Urysohn's lemma and the outer regularity of Radon measures, we get that $\sigma \le \hat{\mu}_{\lfloor E_f}$ by the inner regularity of Radon measures on $\mathcal{R}$, see [54, Theorem 2.18]. Altogether, we deduce that $\sigma = \hat{\mu}_{\lfloor E_f}$, and consequently the measures $\hat{\mu}_{n'_m \lfloor \mathcal{R} \setminus \Omega_m}$ converge weak* to $\hat{\mu}_{\lfloor E_f}$ along $\mathcal{N}' := \{n'_m\}$. In particular,

$$ (3.54) \qquad \lim_{m \to \infty} g(\hat{\mu}_{n'_m \lfloor \mathcal{R} \setminus \Omega_m}, \mathcal{R}; z) = g(\hat{\mu}_{\lfloor E_f}, \mathcal{R}; z), \quad z \in \mathcal{R} \setminus E_f, $$

which settles the asymptotic behavior of the last term in (3.53) along the sequence of indices $\{n'_m\}_m$. Notice that by the definition of $n'_m$, if $\mathcal{N}'$ is replaced by an arbitrary subsequence $\{n''_m\}_m$ thereof, then (3.54) continues to hold along this new subsequence of indices. This will be used in the forthcoming steps.

Next, let $f_m(z, w) := g_{\mathcal{R}}(z, w) - g_{\Omega_m}(z, w)$ for $z, w \in \Omega_m$. Clearly, $f_m(z, \cdot)$ is harmonic in $\Omega_m$ and continuous on $\bar{\Omega}_m$, by the regularity of $\Omega_m$. Its boundary values are equal to $g_{\mathcal{R}}(z, \cdot)$ on $\partial\Omega_m$, in particular they are identically zero on $\partial\mathcal{R}$. Fix $z \in \mathcal{R} \setminus E_f$, and let $k$ be an integer such that $z \in \Omega_k$. Then, we get for all $m > k$ that

$$ (3.55) \qquad 0 \le f_m(z, w) \le \max_{w \in \partial \Omega_m \setminus \partial \mathcal{R}} g_{\mathcal{R}}(z, w) \le \max_{w \in \partial \Omega_{k+1}} g_{\mathcal{R}}(z, w) :=: C_k, \quad w \in \bar{\Omega}_m, $$

where the constant $C_k$ is finite, independent of $m$, and we used the maximum principle for harmonic functions twice (once for $f_m(z, \cdot)$ on $\Omega_m$ and once for $g_{\mathcal{R}}(z, \cdot)$ on $\mathcal{R} \setminus \Omega_{k+1}$). Observe further that the functions $f_m(z, \cdot)$, $m > k$, are not only positive harmonic in each $\Omega_l$ for fixed $l$ satisfying $k < l \le m$, but form a decreasing sequence there. Therefore they converge in $\Omega_l$ to a non-negative harmonic function, say $f^{\{l\}}(z, \cdot)$, by Harnack's theorem. As this claim is true for all large $l$, the $f^{\{l\}}(z, \cdot)$ inductively define a harmonic function $f(z, \cdot)$ on $\mathcal{R} \setminus E_f$ that satisfies $0 \le f(z, \cdot) \le C_k$ when $z \in \Omega_k$. Thus, it extends harmonically to the entire surface $\mathcal{R}$ by the Removability theorem for harmonic functions, and as its trace on $\partial\mathcal{R}$ is zero we conclude that $f(z, \cdot) = 0$.

Observe now that $f_m(z, w) = f_m(w, z)$ for $m$ large enough so that $z, w \in \Omega_m$. Thus, it is jointly harmonic in both variables [35, p. 561]. Hence, by Harnack's theorem and the diagonal argument, any subsequence of $\{f_m(\cdot, \cdot)\}$ has a further subsequence converging locally uniformly in $\mathcal{R} \setminus E_f \times \mathcal{R} \setminus E_f$, and we know from what precedes that the limit function can only be zero. In particular, it holds that

$$ (3.56) \qquad \eta_{l,m} := \max_{z,w \in \Omega_l} f_m(z,w) \to 0 \quad \text{as} \quad m \to \infty, $$

where we used that each $f_m(\cdot, \cdot)$ extends continuously by zero to $\partial\mathcal{R} \times \partial\mathcal{R}$, and therefore the maximum principle for harmonic functions can be applied to show that the convergence is indeed uniform on $\bar{\Omega}_l$. Given $m$, define

$$ l_m := \max \{l : 1 \le l \le m, \eta_{l,m} \le 1/l\}. $$

It follows from (3.56) that $l_m \to \infty$ and $\eta_{l_m, m} \to 0$ when $m \to \infty$, whence for each fixed $z \in \Omega_k$ and all $m > k$ it holds that

$$ (3.57) \qquad 0 \le \int_{\Omega_m} f_m(z, w) d\hat{\mu}_n(w) = \left( \int_{\Omega_m \setminus \Omega_{l_m}} + \int_{\Omega_{l_m}} \right) f_m(z, w) d\hat{\mu}_n(w) \\ \le C_k \hat{\mu}_n(\bar{\Omega}_m \setminus \Omega_{l_m}) + M \eta_{l_m, m}, $$

where we used (3.55) while recalling that $M$ is the number of sheets of $\mathcal{R}$, which is a bound for the total mass of each $\hat{\mu}_n$. Now, we obtain from Lemma 3.4 and our choice of $\Omega_m$ that the measures $\hat{\mu}_n$ converge weak* to $\hat{\mu}$ along $\mathcal{N}$, not only on $\mathcal{R}$, but also on every compact set of the form $\bar{\Omega}_m \setminus \Omega_{l_m}$. Hence, taking into account that $\hat{\mu}((\mathcal{R} \setminus E_f) \setminus \Omega_{l_m}) \to 0$ as $m \to \infty$, we can associate to each $m$ an integer $n''_m \in \mathcal{N}'$ such that $n''_m \ge n'_m$ and

$$ \hat{\mu}_{n''_m}(\bar{\Omega}_m \setminus \Omega_{l_m}) \to 0 \quad \text{as} \quad m \to \infty. $$
Clearly, the choice of $N'' := \{n_m''\}$ is independent of $z$. Thus, we get from (3.57) together with the choices of $\{l_m\}$ and $N''$ that for every $z \in \mathbb{R} \setminus E_f$ it holds

$$ (3.58) \quad \lim_{m \to \infty} g(\hat{\mu}_{n_m''} \lfloor \Omega_m, \mathcal{R}; z) - g(\hat{\mu}_{n_m''}, \Omega_m; z) = \lim_{m \to \infty} \int_{\Omega_m} f_m(z, w) d\hat{\mu}_{n_m''}(w) = 0, $$

which settles the asymptotic behavior of the middle term in (3.53) along $N''$.

Lastly, to describe asymptotics of the first term in (3.53), we need to repeat some steps of the proof of Lemma 3.5 with $v_n$ replaced by $v_n + \hat{\mu}_n$. Recall the definition of the sets $\Omega_{m,l}$ given just before (3.31). We may adjust it so that $(v^* + \hat{\mu})(\partial \Omega_{m,l}) = 0$. Indeed, $\partial \Omega_{m,l} = (\partial \Omega_m \setminus \partial \mathcal{R}) \cup p^{-1}(\mathbb{T}_{r_l})$. We already know that $(v^* + \hat{\mu})(\partial \Omega_m \setminus \partial \mathcal{R}) = 0$, so we only need to ensure that $(v^* + \hat{\mu})(p^{-1}(\mathbb{T}_{r_l})) = 0$ for each $l$. This can be achieved as before since $\hat{\mu}$ is finite and therefore $v^* + \hat{\mu}$ is still a Radon measure. Since $\hat{\mu}'(\partial \Omega_{m,l}) = \hat{\mu}(\partial \Omega_{m,l}) = 0$ by construction, it follows from Lemma 3.4 that

$$ \hat{\mu}_{n \lfloor \bar{\Omega}_m \setminus \Omega_{m,l}} \xrightarrow{w^*} \hat{\mu}'_{\lfloor \bar{\Omega}_m \setminus \Omega_{m,l}} \quad \text{and} \quad \hat{\mu}_n(p^{-1}(\mathbb{T}_{r_l})) \to 0 $$

for all $m, l$. So, when $z \in \Omega_{m,l}$, we get that

$$ (3.59) \quad \lim_{N'' \ni n \to \infty} g(\hat{\mu}_{n \lfloor \Omega_m \setminus \bar{\Omega}_{m,l}}, \Omega_m; z) = \lim_{N'' \ni n \to \infty} g(\hat{\mu}_{n \lfloor \bar{\Omega}_m \setminus \Omega_{m,l}}, \Omega_m; z) \\ = g(\hat{\mu}'_{\lfloor \bar{\Omega}_m \setminus \Omega_{m,l}}, \Omega_m; z) = g(\hat{\mu}_{\lfloor \Omega_m \setminus \bar{\Omega}_{m,l}}, \Omega_m; z) $$

since $g_{\Omega_m}(\cdot, z)$ is continuous on $\bar{\Omega}_m \setminus \Omega_{m,l}$ and vanishes on $\partial \mathcal{R}$ by regularity of $\Omega_m$. Moreover, the convergence is locally uniform in $z \in \Omega_{m,l}$ by the continuity of Green functions with respect to both variables off the diagonal. From (3.59) and the reasoning used after (3.31), it follows that

$$ \lim_{N'' \ni n \to \infty} g((v_n + \hat{\mu}_n)_{\lfloor \Omega_m \setminus \bar{\Omega}_{m,l}}, \Omega_m; z) = h_{m,l}(z) + g(\hat{\mu}_{\lfloor \Omega_m \setminus \bar{\Omega}_{m,l}}, \Omega_m; z), $$

locally uniformly in $\Omega_{m,l}$ for each $l$ and some subsequence $N_m''$ of $N''$. Arguing as we did to obtain (3.32), but applying this time the *Lower Envelope theorem* to $(v_n + \hat{\mu}_n)_{\lfloor \bar{\Omega}_{m,l}}$, we get in view of the above limit that

$$ \liminf_{\tilde{N}'' \ni n \to \infty} g(v_n + \hat{\mu}_n, \Omega_m; z) = g((v^* + \hat{\mu})_{\lfloor \bar{\Omega}_{m,l}}, \Omega_m; z) + h_{m,l}(z) + g(\hat{\mu}_{\lfloor \Omega_m \setminus \bar{\Omega}_{m,l}}, \Omega_m; z) \\ = g(v^* + \hat{\mu}, \Omega_m; z) + h_m(z) $$

for quasi every $z \in \Omega_m$, where $\tilde{N}''$ is the diagonal of the table $\{N_m''\}_{m=1}^{\infty}$ and to get the second equality we used (3.33) along with the explanation preceding it on the inductive definition of $u_m$ by the right-hand side of (3.32). The previous equation stands analog to (3.34), and continues to hold if $\tilde{N}''$ is replaced by any subsequence thereof. Now, the last part of the proof in Lemma 3.5 was predicated on the limit

$$ u_m(z) = g(v^*, \Omega_m; z) + h_m(z) \to u''(z) \quad \text{as} \quad m \to \infty, \ z \in \mathbb{R} \setminus E_f, $$

where $u_m$ were initially defined inductively by the right-hand side of (3.32) and their limit $u''$ assumed the form (3.35). In the present case, $u_m$ is replaced by $u_m + g(\hat{\mu}, \Omega_m; \cdot)$ and the monotone convergence theorem together with the polar character of $E_f$ imply that

$$ \lim_m g(\hat{\mu}, \Omega_m; z) = g(\hat{\mu}, \mathbb{R} \setminus E_f; z) = g(\hat{\mu}_{\lfloor \mathbb{R} \setminus E_f}, \mathcal{R}; z), \quad z \in \mathbb{R} \setminus E_f. $$

Hence, arguing as we did after (3.39), we obtain similarly to (3.43) that

$$ (3.60) \quad \liminf_{m \to \infty} g(v_{n_m}^* + \hat{\mu}_{n_m}^*, \Omega_m; z) = u''(z) + g(\hat{\mu}_{\lfloor \mathbb{R} \setminus E_f}, \mathcal{R}; z), $$

for quasi every $z \in \mathbb{R} \setminus E_f$ and some subsequence $\{n_m^*\} \subset \tilde{N}''$. The desired limit (3.51) now follows from (3.52), (3.53), (3.54), (3.58), and (3.60). $\square$
Clearly, the choice of $N'' := \{n_m''\}$ is independent of $z$. Thus, we get from (3.57) together with the choices of $\{l_m\}$ and $N''$ that for every $z \in \mathbb{R} \setminus E_f$ it holds

$$ (3.58) \quad \lim_{m \to \infty} g(\hat{\mu}_{n_m''} \lfloor \Omega_m, \mathcal{R}; z) - g(\hat{\mu}_{n_m''}, \Omega_m; z) = \lim_{m \to \infty} \int_{\Omega_m} f_m(z, w) d\hat{\mu}_{n_m''}(w) = 0, $$

which settles the asymptotic behavior of the middle term in (3.53) along $N''$.

Lastly, to describe asymptotics of the first term in (3.53), we need to repeat some steps of the proof of Lemma 3.5 with $v_n$ replaced by $v_n + \hat{\mu}_n$. Recall the definition of the sets $\Omega_{m,l}$ given just before (3.31). We may adjust it so that $(v^* + \hat{\mu})(\partial \Omega_{m,l}) = 0$. Indeed, $\partial \Omega_{m,l} = (\partial \Omega_m \setminus \partial \mathcal{R}) \cup p^{-1}(\mathbb{T}_{r_l})$. We already know that $(v^* + \hat{\mu})(\partial \Omega_m \setminus \partial \mathcal{R}) = 0$, so we only need to ensure that $(v^* + \hat{\mu})(p^{-1}(\mathbb{T}_{r_l})) = 0$ for each $l$. This can be achieved as before since $\hat{\mu}$ is finite and therefore $v^* + \hat{\mu}$ is still a Radon measure. Since $\hat{\mu}'(\partial \Omega_{m,l}) = \hat{\mu}(\partial \Omega_{m,l}) = 0$ by construction, it follows from Lemma 3.4 that

$$ \hat{\mu}_{n \lfloor \bar{\Omega}_m \setminus \Omega_{m,l}} \xrightarrow{w^*} \hat{\mu}'_{\lfloor \bar{\Omega}_m \setminus \Omega_{m,l}} \quad \text{and} \quad \hat{\mu}_n(p^{-1}(\mathbb{T}_{r_l})) \to 0 $$

for all $m, l$. So, when $z \in \Omega_{m,l}$, we get that

$$ (3.59) \quad \lim_{N'' \ni n \to \infty} g(\hat{\mu}_{n \lfloor \Omega_m \setminus \bar{\Omega}_{m,l}}, \Omega_m; z) = \lim_{N'' \ni n \to \infty} g(\hat{\mu}_{n \lfloor \bar{\Omega}_m \setminus \Omega_{m,l}}, \Omega_m; z) \\ = g(\hat{\mu}'_{\lfloor \bar{\Omega}_m \setminus \Omega_{m,l}}, \Omega_m; z) = g(\hat{\mu}_{\lfloor \Omega_m \setminus \bar{\Omega}_{m,l}}, \Omega_m; z) $$

since $g_{\Omega_m}(\cdot, z)$ is continuous on $\bar{\Omega}_m \setminus \Omega_{m,l}$ and vanishes on $\partial \mathcal{R}$ by regularity of $\Omega_m$. Moreover, the convergence is locally uniform in $z \in \Omega_{m,l}$ by the continuity of Green functions with respect to both variables off the diagonal. From (3.59) and the reasoning used after (3.31), it follows that

$$ \lim_{N'' \ni n \to \infty} g((v_n + \hat{\mu}_n)_{\lfloor \Omega_m \setminus \bar{\Omega}_{m,l}}, \Omega_m; z) = h_{m,l}(z) + g(\hat{\mu}_{\lfloor \Omega_m \setminus \bar{\Omega}_{m,l}}, \Omega_m; z), $$

locally uniformly in $\Omega_{m,l}$ for each $l$ and some subsequence $N_m''$ of $N''$. Arguing as we did to obtain (3.32), but applying this time the *Lower Envelope theorem* to $(v_n + \hat{\mu}_n)_{\lfloor \bar{\Omega}_{m,l}}$, we get in view of the above limit that

$$ \liminf_{\tilde{N}'' \ni n \to \infty} g(v_n + \hat{\mu}_n, \Omega_m; z) = g((v^* + \hat{\mu})_{\lfloor \bar{\Omega}_{m,l}}, \Omega_m; z) + h_{m,l}(z) + g(\hat{\mu}_{\lfloor \Omega_m \setminus \bar{\Omega}_{m,l}}, \Omega_m; z) \\ = g(v^* + \hat{\mu}, \Omega_m; z) + h_m(z) $$

for quasi every $z \in \Omega_m$, where $\tilde{N}''$ is the diagonal of the table $\{N_m''\}_{m=1}^{\infty}$ and to get the second equality we used (3.33) along with the explanation preceding it on the inductive definition of $u_m$ by the right-hand side of (3.32). The previous equation stands analog to (3.34), and continues to hold if $\tilde{N}''$ is replaced by any subsequence thereof. Now, the last part of the proof in Lemma 3.5 was predicated on the limit

$$ u_m(z) = g(v^*, \Omega_m; z) + h_m(z) \to u''(z) \quad \text{as} \quad m \to \infty, \ z \in \mathbb{R} \setminus E_f, $$

where $u_m$ were initially defined inductively by the right-hand side of (3.32) and their limit $u''$ assumed the form (3.35). In the present case, $u_m$ is replaced by $u_m + g(\hat{\mu}, \Omega_m; \cdot)$ and the monotone convergence theorem together with the polar character of $E_f$ imply that

$$ \lim_m g(\hat{\mu}, \Omega_m; z) = g(\hat{\mu}, \mathbb{R} \setminus E_f; z) = g(\hat{\mu}_{\lfloor \mathbb{R} \setminus E_f}, \mathcal{R}; z), \quad z \in \mathbb{R} \setminus E_f. $$

Hence, arguing as we did after (3.39), we obtain similarly to (3.43) that

$$ (3.60) \quad \liminf_{m \to \infty} g(v_{n_m}^* + \hat{\mu}_{n_m}^*, \Omega_m; z) = u''(z) + g(\hat{\mu}_{\lfloor \mathbb{R} \setminus E_f}, \mathcal{R}; z), $$

for quasi every $z \in \mathbb{R} \setminus E_f$ and some subsequence $\{n_m^*\} \subset \tilde{N}''$. The desired limit (3.51) now follows from (3.52), (3.53), (3.54), (3.58), and (3.60). $\square$
Hereafter we shall deal with the *fine topology*, which is the coarsest topology for which super-harmonic and therefore $\delta$-subharmonic functions are continuous. In this connection, the reader may want to consult the definitions and properties collected in Section A.5.

Recall the definition of $N_0 \subset \mathcal{R}$ before (3.44) as being the $+\infty$-set of $g(\mu, \mathbb{D}; p(\cdot))$, i.e., $N_0 = p^{-1}(\tilde{N}_0)$ where $\tilde{N}_0 \subset \mathbb{D}$ is the $+\infty$-set of $g(\mu, \mathbb{D}; \cdot)$. Clearly, $N_0$ is a finely closed and polar $G_\delta$-set. Let us define

$$ (3.61) \qquad G_{\pm} := \{z \in \mathcal{R} \setminus N_0 : \pm \operatorname{ler}(z) > 0\}. $$

It is easily seen from (3.44) that $\operatorname{ler} : \mathcal{R} \setminus N_0 \to [-\infty, +\infty)$ is finely continuous and that $G_-$ and $G_+$ are finely open in $\mathcal{R} \setminus N_0$. Since $N_0$ is polar, if the complement of either $G_+$ or $G_-$ in $\mathcal{R} \setminus N_0$ is thin at a point $z$, then the respective complement in $\mathcal{R}$ is also thin at $z$. Hence, $G_+$ and $G_-$ are in fact finely open in $\mathcal{R}$. Hereafter, we put $D_\pm := p(G_\pm)$.

**Lemma 3.10.** For $z \in G_+$ and any $\zeta \in \mathcal{R}$ with $p(\zeta) = p(z)$, it holds that $\zeta \in G_+$ and $\operatorname{ler}(z) = \operatorname{ler}(\zeta)$. In particular $G_+ = p^{-1}(D_+)$. Moreover, $D_+$ is finely open.

*Proof.* By the definition of $\operatorname{ler}$ given in (3.46), we get from equation (3.11) together with Lemmas 3.1, 3.7, and 3.9 (recall $N''$ was renamed as $N$ at the top of Section 3.7) that there exists a polar set $B \subset \mathcal{R}$ with the following property: for each $z \in G_+ \setminus B$ there is a subsequence $N_z \subset N$ such that

$$ (3.62) \qquad \left\{ \begin{array}{l} \lim_{N_z \ni n \to \infty} \frac{1}{n} \log |(f - N(M_n) \circ p)(z)| = \operatorname{ler}(z), \\ \\ \lim_{N_z \ni n \to \infty} g(\mu_n, \mathbb{D}; p(z)) = g(\mu, \mathbb{D}; p(z)), \end{array} \right. $$

where we note that $g(\mu, \mathbb{D}; p(z)) < +\infty$ since $z \notin N_0$. Without loss of generality, we may assume that $B$ contains $p^{-1}(p(E_f))$ and $A_0$, since both sets are polar, see Lemma 3.6 for the definition of $A_0$. As $\operatorname{ler}(z) > 0$ and $f(z)$ is finite on $G_+ \setminus B$, the above limit implies that

$$ \lim_{N_z \ni n \to \infty} \frac{1}{n} \log |N(M_n)(p(z))| = \operatorname{ler}(z). $$

Now, if $\zeta \in \mathcal{R} \setminus B$ is such that $p(\zeta) = p(z)$, obviously $\zeta \notin N_0$ and in addition $\zeta \notin E_f$, by definition of $B$. Thus, on account of the finiteness of $f(\zeta)$, we get that

$$ (3.63) \qquad \lim_{N_z \ni n \to \infty} \frac{1}{n} \log |(f - N(M_n) \circ p)(\zeta)| = \lim_{N_z \ni n \to \infty} \frac{1}{n} \log |N(M_n)(p(z))| = \operatorname{ler}(z). $$

On the other hand, from the second equation in (3.62), we deduce as in (3.45) that the first limit in (3.63) is at most $\operatorname{ler}(\zeta)$, whence $0 < \operatorname{ler}(z) \le \operatorname{ler}(\zeta)$. In particular, $\zeta \in G_+$, and reversing the roles of $z$ and $\zeta$ gives $\operatorname{ler}(z) = \operatorname{ler}(\zeta)$. This proves the first assertion of the lemma when $z \in G_+ \setminus B$.

To prove it on all of $G_+$, pick $z$, $\zeta \in \mathcal{R}$ such that $p(z) = p(\zeta)$ and let $D_0 \subset \mathbb{D}$ be a disk centered at $p(z)$. Denote by $D_z$, $D_\zeta$ the connected components of $p^{-1}(D_0)$ that contain $z$, $\zeta$ respectively, and make $D_0$ small enough so that $D_z \cap \operatorname{rp}(\mathcal{R}) \subseteq \{z\}$ and $D_\zeta \cap \operatorname{rp}(\mathcal{R}) \subseteq \{\zeta\}$. Let as before $m(\xi)$ be the ramification order of $\xi$, so that $D_z$ (resp. $D_\zeta$) is (isomorphic to) an $m(z)$-sheeted cyclic covering of $D_0$. For $x \in D_0 \setminus \tilde{N}_0$, define

$$ g(x) := m(\zeta) \sum_{\xi \in p^{-1}(x) \cap D_z} m(\xi) \operatorname{ler}(\xi) - m(z) \sum_{\xi \in p^{-1}(x) \cap D_\zeta} m(\xi) \operatorname{ler}(\xi). $$

It follows from (3.46) and (A.32) that

$$ \sum_{\xi \in p^{-1}(x) \cap D_z} m(\xi) \operatorname{ler}(\xi) = m(z)g(\mu, \mathbb{D}; x) - g(p_*(v_{\lfloor D_z}), \mathbb{D}; x) \\ - \sum_{\xi \in p^{-1}(x) \cap D_z} m(\xi) (g(v_{\lfloor \mathcal{R} \setminus D_z}, \mathcal{R}; \xi) + h_{\mathcal{R}}(\xi)). $$
Hereafter we shall deal with the *fine topology*, which is the coarsest topology for which super-harmonic and therefore $\delta$-subharmonic functions are continuous. In this connection, the reader may want to consult the definitions and properties collected in Section A.5.

Recall the definition of $N_0 \subset \mathcal{R}$ before (3.44) as being the $+\infty$-set of $g(\mu, \mathbb{D}; p(\cdot))$, i.e., $N_0 = p^{-1}(\tilde{N}_0)$ where $\tilde{N}_0 \subset \mathbb{D}$ is the $+\infty$-set of $g(\mu, \mathbb{D}; \cdot)$. Clearly, $N_0$ is a finely closed and polar $G_\delta$-set. Let us define

$$ (3.61) \qquad G_{\pm} := \{z \in \mathcal{R} \setminus N_0 : \pm \operatorname{ler}(z) > 0\}. $$

It is easily seen from (3.44) that $\operatorname{ler} : \mathcal{R} \setminus N_0 \to [-\infty, +\infty)$ is finely continuous and that $G_-$ and $G_+$ are finely open in $\mathcal{R} \setminus N_0$. Since $N_0$ is polar, if the complement of either $G_+$ or $G_-$ in $\mathcal{R} \setminus N_0$ is thin at a point $z$, then the respective complement in $\mathcal{R}$ is also thin at $z$. Hence, $G_+$ and $G_-$ are in fact finely open in $\mathcal{R}$. Hereafter, we put $D_\pm := p(G_\pm)$.

**Lemma 3.10.** For $z \in G_+$ and any $\zeta \in \mathcal{R}$ with $p(\zeta) = p(z)$, it holds that $\zeta \in G_+$ and $\operatorname{ler}(z) = \operatorname{ler}(\zeta)$. In particular $G_+ = p^{-1}(D_+)$. Moreover, $D_+$ is finely open.

*Proof.* By the definition of $\operatorname{ler}$ given in (3.46), we get from equation (3.11) together with Lemmas 3.1, 3.7, and 3.9 (recall $N''$ was renamed as $N$ at the top of Section 3.7) that there exists a polar set $B \subset \mathcal{R}$ with the following property: for each $z \in G_+ \setminus B$ there is a subsequence $N_z \subset N$ such that

$$ (3.62) \qquad \left\{ \begin{array}{l} \lim_{N_z \ni n \to \infty} \frac{1}{n} \log |(f - N(M_n) \circ p)(z)| = \operatorname{ler}(z), \\ \lim_{N_z \ni n \to \infty} g(\mu_n, \mathbb{D}; p(z)) = g(\mu, \mathbb{D}; p(z)), \end{array} \right. $$

where we note that $g(\mu, \mathbb{D}; p(z)) < +\infty$ since $z \notin N_0$. Without loss of generality, we may assume that $B$ contains $p^{-1}(p(E_f))$ and $A_0$, since both sets are polar, see Lemma 3.6 for the definition of $A_0$. As $\operatorname{ler}(z) > 0$ and $f(z)$ is finite on $G_+ \setminus B$, the above limit implies that

$$ \lim_{N_z \ni n \to \infty} \frac{1}{n} \log |N(M_n)(p(z))| = \operatorname{ler}(z). $$

Now, if $\zeta \in \mathcal{R} \setminus B$ is such that $p(\zeta) = p(z)$, obviously $\zeta \notin N_0$ and in addition $\zeta \notin E_f$, by definition of $B$. Thus, on account of the finiteness of $f(\zeta)$, we get that

$$ (3.63) \qquad \lim_{N_z \ni n \to \infty} \frac{1}{n} \log |(f - N(M_n) \circ p)(\zeta)| = \lim_{N_z \ni n \to \infty} \frac{1}{n} \log |N(M_n)(p(z))| = \operatorname{ler}(z). $$

On the other hand, from the second equation in (3.62), we deduce as in (3.45) that the first limit in (3.63) is at most $\operatorname{ler}(\zeta)$, whence $0 < \operatorname{ler}(z) \le \operatorname{ler}(\zeta)$. In particular, $\zeta \in G_+$, and reversing the roles of $z$ and $\zeta$ gives $\operatorname{ler}(z) = \operatorname{ler}(\zeta)$. This proves the first assertion of the lemma when $z \in G_+ \setminus B$.

To prove it on all of $G_+$, pick $z$, $\zeta \in \mathcal{R}$ such that $p(z) = p(\zeta)$ and let $D_0 \subset \mathbb{D}$ be a disk centered at $p(z)$. Denote by $D_z$, $D_\zeta$ the connected components of $p^{-1}(D_0)$ that contain $z$, $\zeta$ respectively, and make $D_0$ small enough so that $D_z \cap \operatorname{rp}(\mathcal{R}) \subseteq \{z\}$ and $D_\zeta \cap \operatorname{rp}(\mathcal{R}) \subseteq \{\zeta\}$. Let as before $m(\xi)$ be the ramification order of $\xi$, so that $D_z$ (resp. $D_\zeta$) is (isomorphic to) an $m(z)$-sheeted cyclic covering of $D_0$. For $x \in D_0 \setminus \tilde{N}_0$, define

$$ g(x) := m(\zeta) \sum_{\xi \in p^{-1}(x) \cap D_z} m(\xi) \operatorname{ler}(\xi) - m(z) \sum_{\xi \in p^{-1}(x) \cap D_\zeta} m(\xi) \operatorname{ler}(\xi). $$

It follows from (3.46) and (A.32) that

$$ \sum_{\xi \in p^{-1}(x) \cap D_z} m(\xi) \operatorname{ler}(\xi) = m(z)g(\mu, \mathbb{D}; x) - g(p_*(v_{\lfloor D_z}), \mathbb{D}; x) \\ - \sum_{\xi \in p^{-1}(x) \cap D_z} m(\xi) \left(g(v_{\lfloor \mathcal{R} \setminus D_z}}, \mathcal{R}; \xi) + h_{\mathcal{R}}(\xi)\right). $$
Notice that the last summand above is a continuous, even harmonic function in $D_0$. Hence, the sum itself is finely continuous in $D_0 \setminus \tilde{N}_0$. Likewise, the second sum in the definition of $g$ is finely continuous in $D_0 \setminus \tilde{N}_0$ and so is $g$. In particular, $U := \{x \in D_0 \setminus \tilde{N}_0 : g(x) \neq 0\}$ is finely open, and since $p(G_+)$ is finely open by Lemma A.2 the set $U \cap p(G_+)$ is in turn finely open. From the first part of the proof, it follows that if $x \in (D_0 \cap p(G_+)) \setminus p(B)$ then $g(x) = 0$, whence $U \cap p(G_+) \subseteq p(B)$. Thus, $U \cap p(G_+)$ must be empty as $p(B)$ is polar, that is, $g \equiv 0$ on $D_0 \cap p(G_+)$. Now, it can be readily checked that

$$g(p(z)) = m(z)m(\zeta)(ler(z) - ler(\zeta)),$$

and therefore $ler(z) = ler(\zeta)$ if $z \in G_+$, thereby proving the first assertion of the lemma. The second is then obvious, and the third follows from Lemma A.2. $\square$

**Lemma 3.11.** The set $G_-$ lies schlicht over $\mathbb{D}$. That is, $p : G_- \to D_-$ is a bijection. Moreover, $G_- \cap \mathbf{rp}(\mathcal{R}) = \emptyset$, and for each $z \in G_-$ we have that

$$(3.64) \qquad ler(\zeta) = 0 \text{ for all } \zeta \in p^{-1}(p(z))\setminus\{z\}.$$

*Proof.* Pick $z \in G_- \setminus \mathbf{rp}(\mathcal{R})$ and let $D_0 \subset \mathcal{R} \setminus \mathbf{rp}(\mathcal{R})$ be a conformal disk centered at $z$, homeomorphic under $p$ to a Euclidean disk $p(D_0)$; note that $p^{-1}(D_0) \setminus D_0$ is open. Define $U := G_- \cap D_0$ and $V := (p^{-1}(D_0) \setminus D_0) \cap p^{-1}(p(U))$. Clearly $U$ and $V$ are disjoint finely open subsets of $\mathcal{R}$, by Lemma A.2. In view of Lemma 3.6, there is a polar set $A_0 \subset \mathcal{R} \setminus N_0$ such that $V \cap G_- \subseteq A_0$. Thus, $V \cap G_-$ must be empty as otherwise it is finely open. This shows that $G_- \setminus \mathbf{rp}(\mathcal{R})$ lies schlicht over $\mathbb{D}$. However, the complement of a schlicht set is always non-thin at any ramification point by Lemma A.4. Hence, $G_-$ cannot be a fine neighborhood of a ramification point, and since it is finely open $G_- \cap \mathbf{rp}(\mathcal{R}) = \emptyset$. Altogether, $G_-$ lies schlicht over $\mathbb{D}$ and (3.64) now follows from this and Lemma 3.10. $\square$

**3.8. Modified Logarithmic Error Function.** In this subsection we modify the function $ler(z)$ by clearing out parts of $\mathbb{D}$ and $\mathcal{R}$ from the support of $\mu$ and $v$, respectively. We shall accomplish this via the technique of balayage, described in Section A.9. Let us start with some preliminary geometric considerations. Recall that any connected (topological) 1-manifold embedded in $\mathcal{R}$ is a Jordan curve.

**Lemma 3.12.** For each $\epsilon > 0$ there exists a Jordan curve $J \subset G_-$ such that $p(J)$ is a Jordan curve included in $\{z : 1 - \epsilon < |z| < 1\}$ and $p(\mathbf{rp}(\mathcal{R}))$ belongs to the interior domain of $p(J)$. Moreover, there exists a finely connected component of $G_-$, say $G_J$, such that $J \subset G_J$.

*Proof.* We may assume that $\epsilon$ is small enough that $p(\mathbf{rp}(\mathcal{R})) \subset D_{1-\epsilon}$. In particular, $p$ is injective on every conformal disk centered at a point of $\partial\mathcal{R}$ with radius smaller than or equal to $\epsilon$.

Recall that $\mathcal{R}$ is a subset of a Riemann surface $\mathcal{R}_*$ lying over $\bar{\mathbb{C}}$. Define $G_*^- := G_- \cup \mathcal{T} \cup \mathcal{D}$, where $\mathcal{D}$ is the connected component of $p^{-1}(\bar{\mathbb{C}} \setminus \bar{\mathbb{D}})$ that borders $\mathcal{T}$. Let us show that for each $\eta \in \mathcal{T}$ there is a disk $D_\eta \subset \{z : 1 - \epsilon < |z| < 1 + \epsilon\}$, centered at $p(\eta)$ with radius $r_\eta$, such that the circle $\partial D_\eta$ is included in $p(G_*^-)$. In fact, we can pick $r_\eta$ so that there exist radii $r'_\eta$ arbitrarily close but not equal to $r_\eta$ for which each disk $D'_\eta$ centered at $p(\eta)$ of radius $r'_\eta$ also satisfies $\partial D'_\eta \subset p(G_*^-)$. Indeed, like we did to establish Lemma 3.9, let $\hat{\mu}$ be the lift $\mu$ to $\mathcal{R}$, see (A.33). By definition, $ler(z) \ge 0$ for $z \in \mathcal{R} \setminus (G_- \cup N_0)$. Therefore, if $\eta \in \mathcal{T}$ is a limit point of $\mathcal{R} \setminus G_-$, then we get from (3.17), (3.46), the definition of $h_{\mathcal{R}}(z)$, and the identity $g(\hat{\mu}, \mathcal{R}; \zeta) = +\infty$ when $\zeta \in N_0$ that

$$\liminf_{\mathcal{R} \setminus G_- \ni \zeta \to \eta} g(\hat{\mu}, \mathcal{R}; \zeta) \ge 2/\mathrm{cap}_{\mathbb{D}}(K_f).$$

The claim now follows from Lemma A.5 by taking $1 - r_\eta$ to be an accumulation point of $R_{1/\mathrm{cap}_{\mathbb{D}}(K_f)}$ in $(1 - \epsilon, 1)$.

Let $U_\eta$ be the connected component of $p^{-1}(D_\eta)$ containing $\eta$, which is an open subset of $\mathcal{R}_*$ satisfying $\partial U_\eta \subset G_*^-$. Since the collection $\{U_\eta\}_\eta$ covers $\mathcal{T}$, which is compact, it contains a finite subcover, say $\{U_{\eta_i}\}$. Replacing $D_{\eta_i}$ by some $D'_{\eta_i}$ as above if needed, we can ensure by
Notice that the last summand above is a continuous, even harmonic function in $D_0$. Hence, the sum itself is finely continuous in $D_0 \setminus \tilde{N}_0$. Likewise, the second sum in the definition of $g$ is finely continuous in $D_0 \setminus \tilde{N}_0$ and so is $g$. In particular, $U := \{x \in D_0 \setminus \tilde{N}_0 : g(x) \neq 0\}$ is finely open, and since $p(G_+)$ is finely open by Lemma A.2 the set $U \cap p(G_+)$ is in turn finely open. From the first part of the proof, it follows that if $x \in (D_0 \cap p(G_+)) \setminus p(B)$ then $g(x) = 0$, whence $U \cap p(G_+) \subseteq p(B)$. Thus, $U \cap p(G_+)$ must be empty as $p(B)$ is polar, that is, $g \equiv 0$ on $D_0 \cap p(G_+)$. Now, it can be readily checked that

$$g(p(z)) = m(z)m(\zeta)(ler(z) - ler(\zeta)),$$

and therefore $ler(z) = ler(\zeta)$ if $z \in G_+$, thereby proving the first assertion of the lemma. The second is then obvious, and the third follows from Lemma A.2. $\square$

**Lemma 3.11.** The set $G_-$ lies schlicht over $\mathbb{D}$. That is, $p : G_- \to D_-$ is a bijection. Moreover, $G_- \cap rp(\mathcal{R}) = \emptyset$, and for each $z \in G_-$ we have that

$$(3.64) \qquad ler(\zeta) = 0 \text{ for all } \zeta \in p^{-1}(p(z))\setminus\{z\}.$$

*Proof.* Pick $z \in G_- \setminus rp(\mathcal{R})$ and let $D_0 \subset \mathcal{R} \setminus rp(\mathcal{R})$ be a conformal disk centered at $z$, homeomorphic under $p$ to a Euclidean disk $p(D_0)$; note that $p^{-1}(D_0) \setminus D_0$ is open. Define $U := G_- \cap D_0$ and $V := (p^{-1}(D_0) \setminus D_0) \cap p^{-1}(p(U))$. Clearly $U$ and $V$ are disjoint finely open subsets of $\mathcal{R}$, by Lemma A.2. In view of Lemma 3.6, there is a polar set $A_0 \subset \mathcal{R} \setminus N_0$ such that $V \cap G_- \subseteq A_0$. Thus, $V \cap G_-$ must be empty as otherwise it is finely open. This shows that $G_- \setminus rp(\mathcal{R})$ lies schlicht over $\mathbb{D}$. However, the complement of a schlicht set is always non-thin at any ramification point by Lemma A.4. Hence, $G_-$ cannot be a fine neighborhood of a ramification point, and since it is finely open $G_- \cap rp(\mathcal{R}) = \emptyset$. Altogether, $G_-$ lies schlicht over $\mathbb{D}$ and (3.64) now follows from this and Lemma 3.10. $\square$

**3.8. Modified Logarithmic Error Function.** In this subsection we modify the function $ler(z)$ by clearing out parts of $\mathbb{D}$ and $\mathcal{R}$ from the support of $\mu$ and $v$, respectively. We shall accomplish this via the technique of balayage, described in Section A.9. Let us start with some preliminary geometric considerations. Recall that any connected (topological) 1-manifold embedded in $\mathcal{R}$ is a Jordan curve.

**Lemma 3.12.** For each $\epsilon > 0$ there exists a Jordan curve $J \subset G_-$ such that $p(J)$ is a Jordan curve included in $\{z : 1 - \epsilon < |z| < 1\}$ and $p rp(\mathcal{R})$ belongs to the interior domain of $p(J)$. Moreover, there exists a finely connected component of $G_-$, say $G_J$, such that $J \subset G_J$.

*Proof.* We may assume that $\epsilon$ is small enough that $p rp(\mathcal{R}) \subset D_{1-\epsilon}$. In particular, $p$ is injective on every conformal disk centered at a point of $\partial \mathcal{R}$ with radius smaller than or equal to $\epsilon$.

Recall that $\mathcal{R}$ is a subset of a Riemann surface $\mathcal{R}_*$ lying over $\bar{\mathbb{C}}$. Define $G_*^- := G_- \cup \mathcal{T} \cup \mathcal{D}$, where $\mathcal{D}$ is the connected component of $p^{-1}(\bar{\mathbb{C}} \setminus \bar{\mathbb{D}})$ that borders $\mathcal{T}$. Let us show that for each $\eta \in \mathcal{T}$ there is a disk $D_\eta \subset \{z : 1 - \epsilon < |z| < 1 + \epsilon\}$, centered at $p(\eta)$ with radius $r_\eta$, such that the circle $\partial D_\eta$ is included in $p(G_*^-)$. In fact, we can pick $r_\eta$ so that there exist radii $r'_\eta$ arbitrarily close but not equal to $r_\eta$ for which each disk $D'_\eta$ centered at $p(\eta)$ of radius $r'_\eta$ also satisfies $\partial D'_\eta \subset p(G_*^-)$. Indeed, like we did to establish Lemma 3.9, let $\hat{\mu}$ be the lift $\mu$ to $\mathcal{R}$, see (A.33). By definition, $ler(z) \ge 0$ for $z \in \mathcal{R} \setminus (G_- \cup N_0)$. Therefore, if $\eta \in \mathcal{T}$ is a limit point of $\mathcal{R} \setminus G_-$, then we get from (3.17), (3.46), the definition of $h_{\mathcal{R}}(z)$, and the identity $g(\hat{\mu}, \mathcal{R}; \zeta) = +\infty$ when $\zeta \in N_0$ that

$$\liminf_{\mathcal{R} \setminus G_- \ni \zeta \to \eta} g(\hat{\mu}, \mathcal{R}; \zeta) \ge 2/cap_{\mathbb{D}}(K_f).$$

The claim now follows from Lemma A.5 by taking $1 - r_\eta$ to be an accumulation point of $R_{1/cap_{\mathbb{D}}(K_f)}$ in $(1 - \epsilon, 1)$.

Let $U_\eta$ be the connected component of $p^{-1}(D_\eta)$ containing $\eta$, which is an open subset of $\mathcal{R}_*$ satisfying $\partial U_\eta \subset G_*^-$. Since the collection $\{U_\eta\}_\eta$ covers $\mathcal{T}$, which is compact, it contains a finite subcover, say $\{U_{\eta_i}\}$. Replacing $D_{\eta_i}$ by some $D'_{\eta_i}$ as above if needed, we can ensure by
(finite) induction on $i$ that $\partial U_{\eta_i}$ and $\partial U_{\eta_i}$ may intersect only transversally for $i \neq j$, and that $\partial U_{\eta_i} \cap \partial U_{\eta_j} \cap \partial U_{\eta_k} = \emptyset$ if $i, j, k$ are all distinct. Then $V := \cup_i U_{\eta_i}$ is an open neighborhood of $\mathcal{T}$ with boundary $\partial V$ included in $G^*$ that consists of a finite union of disjoint Lipschitz-smooth Jordan curves. In fact, there are exactly two such curves, one in $\mathcal{R}$ and $\mathcal{D}$, because each connected component of $\partial V$ can be continuously deformed into $\mathcal{T}$ via radial retraction within $V$. We can choose the component within $\mathcal{R}$ to be $J$ since Lipschitz curves are finely connected in $\mathcal{R}$, see Section A.5. $\square$

Recall now the finely open subsets $D_+$ and $D_-$ of $\mathcal{D}$ introduced before Lemma 3.10. Denote by $D_J$ the union of all finely connected components of $D_+ \cup D_-$ that lie entirely within the interior domain of $p(J)$, where $J$ was introduced in Lemma 3.12. Note that $D_J$ is finely open because so are the fine components of finely open sets, see Section A.5. Define

$$ (3.65) \qquad D' := D_J \cup i(\mathcal{D}\setminus D_J) \quad \text{and} \quad V' := p^{-1}(D'), $$

where $i(\cdot)$ is the subset of finely isolated points. Thus, it follows from (A.9) and Lemma A.2 that $D'$ and $V'$ are finely open while

$$ (3.66) \qquad b(\mathcal{D}\setminus D') = \mathcal{D}\setminus D' \quad \text{and} \quad b(\mathcal{R}\setminus V') = \mathcal{R}\setminus V', $$

where $b(\cdot)$ stands for the base of a set (in particular, $\mathcal{D}\setminus D'$ has no finely isolated points). In other words, $D'$ and $V'$ are regular finely open sets, see Section A.7. Recall from Section A.9 the notation $\sigma^E$ for the balayage of the measure $\sigma$ onto the set $E$.

**Lemma 3.13.** Let $N_1 := p^{-1}(\tilde{N}_1)$, where $\tilde{N}_1$ is the $+\infty$-set of $g(\mu^{(1)}, \mathcal{D}; \cdot)$ and we set

$$ (3.67) \qquad \mu^{(1)} := \mu^{\mathcal{D}\setminus D'} \quad \text{and} \quad v^{(1)} := v^{\mathcal{R}\setminus V'}. $$

Then $N_1 \subseteq N_0$, $N_1\setminus V' = N_0\setminus V'$, and for every $z \in \mathcal{R}\setminus N_1$ we can define

$$ \mathrm{ler}^{(1)}(z) := g(\mu^{(1)}, \mathcal{D}; p(z)) - g(v^{(1)}, \mathcal{R}; z) - h_{\mathcal{R}}(z) $$

with values in $(-\infty, +\infty)$. This function satisfies

$$ (3.68) \qquad \mathrm{ler}^{(1)}(z) = \begin{cases} \mathrm{ler}(z), & z \in \mathcal{R}\setminus(V' \cup N_1), \\ 0, & z \in V'\setminus N_1. \end{cases} $$

*Proof.* Since the Green potential of a measure dominates the Green potential of any balayage of that measure, as explained at the beginning of Section A.9, it holds that

$$ g(\mu^{(1)}, \mathcal{D}; z) \le g(\mu, \mathcal{D}; z) < +\infty, \quad z \in \mathcal{R}\setminus N_0, $$

by the very definition of $N_0$. Thus, $N_1 \subseteq N_0$ and the upper equality in (3.68) as well as equality $N_1\setminus V' = N_0\setminus V'$ are consequences of (3.66) and (A.22). Since $h_{\mathcal{R}}(z)$ is a harmonic function on $\mathcal{R}$ and $\overline{V'} \cap \partial \mathcal{R} = \emptyset$ by construction (recall that $p(V')$ lies interior to $p(J)$), equation (A.39) yields that

$$ (3.69) \qquad \int h_{\mathcal{R}}(x) d\delta_z^{\mathcal{R}\setminus V'}(x) = h_{\mathcal{R}}(z), \quad z \in V'. $$

Since $V'$ is a regular finely open set, Lemma A.6 implies that $\delta_z^{\mathcal{R}\setminus V'}$ is carried by $\partial_t V'$ and does not charge polar sets. As $\mathrm{ler}(z) = 0$ for quasi every $z \in \partial_t V'$ by the definition of $V'$, we get from the definition of $\mathrm{ler}^{(1)}$, (3.69), (A.25), Lemma A.7, and (3.46) that

$$ \begin{align*} \mathrm{ler}^{(1)}(z) &= \int (g(\mu, \mathcal{D}; p(x)) - g(v, \mathcal{R}; x) - h_{\mathcal{R}}(x)) d\delta_z^{\mathcal{R}\setminus V'}(x) \\ &= \int \mathrm{ler}(x) d\delta_z^{\mathcal{R}\setminus V'}(x) = 0 \end{align*} $$

for $z \in V'\setminus N_1$, which proves the lower equality in (3.68). $\square$
(finite) induction on $i$ that $\partial U_{\eta_i}$ and $\partial U_{\eta_i}$ may intersect only transversally for $i \neq j$, and that $\partial U_{\eta_i} \cap \partial U_{\eta_j} \cap \partial U_{\eta_k} = \emptyset$ if $i, j, k$ are all distinct. Then $V := \cup_i U_{\eta_i}$ is an open neighborhood of $\mathcal{T}$ with boundary $\partial V$ included in $G^*$ that consists of a finite union of disjoint Lipschitz-smooth Jordan curves. In fact, there are exactly two such curves, one in $\mathcal{R}$ and $\mathcal{D}$, because each connected component of $\partial V$ can be continuously deformed into $\mathcal{T}$ via radial retraction within $V$. We can choose the component within $\mathcal{R}$ to be $J$ since Lipschitz curves are finely connected in $\mathcal{R}$, see Section A.5. $\square$

Recall now the finely open subsets $D_+$ and $D_-$ of $\mathbb{D}$ introduced before Lemma 3.10. Denote by $D_J$ the union of all finely connected components of $D_+ \cup D_-$ that lie entirely within the interior domain of $p(J)$, where $J$ was introduced in Lemma 3.12. Note that $D_J$ is finely open because so are the fine components of finely open sets, see Section A.5. Define

$$ (3.65) \qquad D' := D_J \cup i(\mathbb{D}\setminus D_J) \quad \text{and} \quad V' := p^{-1}(D'), $$

where $i(\cdot)$ is the subset of finely isolated points. Thus, it follows from (A.9) and Lemma A.2 that $D'$ and $V'$ are finely open while

$$ (3.66) \qquad b(\mathbb{D}\setminus D') = \mathbb{D}\setminus D' \quad \text{and} \quad b(\mathcal{R}\setminus V') = \mathcal{R}\setminus V', $$

where $b(\cdot)$ stands for the base of a set (in particular, $\mathbb{D}\setminus D'$ has no finely isolated points). In other words, $D'$ and $V'$ are regular finely open sets, see Section A.7. Recall from Section A.9 the notation $\sigma^E$ for the balayage of the measure $\sigma$ onto the set $E$.

**Lemma 3.13.** Let $N_1 := p^{-1}(\tilde{N}_1)$, where $\tilde{N}_1$ is the $+\infty$-set of $g(\mu^{(1)}, \mathbb{D}; \cdot)$ and we set

$$ (3.67) \qquad \mu^{(1)} := \mu^{\mathbb{D}\setminus D'} \quad \text{and} \quad v^{(1)} := v^{\mathcal{R}\setminus V'}. $$

Then $N_1 \subseteq N_0$, $N_1 \setminus V' = N_0 \setminus V'$, and for every $z \in \mathcal{R}\setminus N_1$ we can define

$$ \mathrm{ler}^{(1)}(z) := g(\mu^{(1)}, \mathbb{D}; p(z)) - g(v^{(1)}, \mathcal{R}; z) - h_{\mathcal{R}}(z) $$

with values in $(-\infty, +\infty)$. This function satisfies

$$ (3.68) \qquad \mathrm{ler}^{(1)}(z) = \begin{cases} \mathrm{ler}(z), & z \in \mathcal{R}\setminus(V' \cup N_1), \\ 0, & z \in V'\setminus N_1. \end{cases} $$

*Proof.* Since the Green potential of a measure dominates the Green potential of any balayage of that measure, as explained at the beginning of Section A.9, it holds that

$$ g(\mu^{(1)}, \mathbb{D}; z) \le g(\mu, \mathbb{D}; z) < +\infty, \quad z \in \mathcal{R}\setminus N_0, $$

by the very definition of $N_0$. Thus, $N_1 \subseteq N_0$ and the upper equality in (3.68) as well as equality $N_1 \setminus V' = N_0 \setminus V'$ are consequences of (3.66) and (A.22). Since $h_{\mathcal{R}}(z)$ is a harmonic function on $\mathcal{R}$ and $\overline{V'} \cap \partial \mathcal{R} = \emptyset$ by construction (recall that $p(V')$ lies interior to $p(J)$), equation (A.39) yields that

$$ (3.69) \qquad \int h_{\mathcal{R}}(x) d\delta_z^{\mathcal{R}\setminus V'}(x) = h_{\mathcal{R}}(z), \quad z \in V'. $$

Since $V'$ is a regular finely open set, Lemma A.6 implies that $\delta_z^{\mathcal{R}\setminus V'}$ is carried by $\partial_t V'$ and does not charge polar sets. As $\mathrm{ler}(z) = 0$ for quasi every $z \in \partial_t V'$ by the definition of $V'$, we get from the definition of $\mathrm{ler}^{(1)}$, (3.69), (A.25), Lemma A.7, and (3.46) that

$$ \begin{align*} \mathrm{ler}^{(1)}(z) &= \int (g(\mu, \mathbb{D}; p(x)) - g(v, \mathcal{R}; x) - h_{\mathcal{R}}(x)) d\delta_z^{\mathcal{R}\setminus V'}(x) \\ &= \int \mathrm{ler}(x) d\delta_z^{\mathcal{R}\setminus V'}(x) = 0 \end{align*} $$

for $z \in V'\setminus N_1$, which proves the lower equality in (3.68). $\square$
Let $G_J$ be as in Lemma 3.12 for some small $\epsilon > 0$. Denote by $G_J^*$ the union of $G_J$ and the annular region delimited by $J$ and $\mathcal{T}$; the latter is diffeomorphic under $p$ to $\{z: 1 - \epsilon < |z| < 1\}$ if we fix $\epsilon$ small enough. Clearly, $G_J^*$ is a fine domain that lies schlicht over $\mathbb{D}$. Let $\mathcal{G}$ be the collection of all fine domains $G \subset \mathbb{R}$ lying schlicht over $\mathbb{D}$ and containing $G_J^*$. The set $\mathcal{G}$ is partially ordered by inclusion and every chain in it is bounded above by the union of its elements. Therefore, by Zorn's lemma, $\mathcal{G}$ possesses a maximal element, say $G_{\max}$. Note that if $\mathbb{R}\setminus G_{\max}$ is thin at two points $\zeta_1, \zeta_2 \in \mathbb{R}$, then $p(\zeta_1) \neq p(\zeta_2)$ as otherwise $G_{\max}$ could not be schlicht over $\mathbb{D}$ by Lemma A.2. Hence, it follows from (A.9) and the maximality of $G_{\max}$ that $\mathbb{R}\setminus G_{\max}$ is its own base.

**Lemma 3.14.** Any maximal domain $G_{\max} \in \mathcal{G}$ is a Euclidean domain. Moreover, no connected component of $\mathbb{R}\setminus G_{\max}$ (resp. $\mathbb{D}\setminus p(G_{\max})$) consists of a single point.

*Proof.* Observe that $G_{\max}$ cannot contain a ramification point of $\mathbb{R}$ as it is finely open and lies schlicht over $\mathbb{D}$, see Lemma A.4. Thus, for every $\zeta \in G_{\max}$ there exists $r_0 > 0$ such that each component of

$$p^{-1}(\{z \in \mathbb{D}: |z - p(\zeta)| < r_0\})$$

is in one-to-one correspondence with $\{z \in \mathbb{D}: |z - p(\zeta)| < r_0\}$ under $p$. Let $V$ be the component containing $\zeta$. Since the intersection $V \cap G_{\max}$ is finely open, $V \setminus G_{\max}$ is thin at $\zeta$ and therefore there is $r_1 \in (0, r_0)$ such that

$$V \cap p^{-1}(\{z \in \mathbb{D}: |z - p(\zeta)| = r_1\}) \subset G_{\max},$$

see Section A.6. Now, as $G_{\max}$ lies schlicht over $\mathbb{D}$, it holds that

$$(3.70) \qquad G_{\max} \cap (p^{-1}(\{z \in \mathbb{D}: |z - p(\zeta)| = r_1\}) \setminus V) = \emptyset.$$

Further, since $G_{\max}$ is finely connected, we necessarily have that

$$G_{\max} \cap (p^{-1}(\{z \in \mathbb{D}: |z - p(\zeta)| < r_1\}) \setminus V) = \emptyset,$$

otherwise $G_{\max}$ would intersect the fine boundary of $p^{-1}(\{z \in \mathbb{D}: |z - p(\zeta)| < r_1\}) \setminus V$ which is contained in $p^{-1}(\{z \in \mathbb{D}: |z - p(\zeta)| = r_1\}) \setminus V$ (in fact, equal to it by regularity), thereby contradicting (3.70). The maximality of $G_{\max}$ now yields that

$$p^{-1}(\{z \in \mathbb{D}: |z - \zeta| < r_1\}) \cap V \subset G_{\max},$$

hence $\zeta$ belongs to the Euclidean interior of $G_{\max}$ and so $G_{\max}$ is Euclidean open. Thus, its connected components are open and therefore finely open. Hence, $G_{\max}$ is a Euclidean domain since it is finely connected by definition.

To prove the second assertion, assume to the contrary that a point $\zeta \in \mathbb{R}\setminus G_{\max}$ is a connected component of the latter. We claim that any open neighborhood $W$ of $\zeta$ contains an open set $O \ni \zeta$ whose boundary $\partial O$ is a smooth Jordan curve $C$ contained in $G_{\max}$. To see this, assume with no loss of generality that $\bar{W} \subset \mathbb{R}$, and put $F := \mathbb{R}\setminus G_{\max} \cap \bar{W}$. The latter is closed in $\mathbb{R}$, so there is a smooth function $h: \mathbb{R} \to \mathbb{R}^+$ of which $F$ is the zero set, see Section 3.4. Given a sequence $\{c_n\}$ of regular values of $h$ tending to 0, let $O_n$ be the connected component containing $\zeta$ of the open set $\{z: h(z) < c_n\}$. The sets $O_n$ form a (strictly) nested sequence of connected open sets containing $\zeta$, whose intersection is connected and contained in $F$ whence reduces to $\zeta$. If $O_{n_k} \cap \partial W \neq \emptyset$ for some increasing subsequence of indices $n_k$, take $z_{n_k} \in O_{n_k} \cap \partial W$ and extract from $\{z_{n_k}\}$ a subsequence converging to $z \in \partial W$, which is possible due to compactness of the latter. On the other hand, since $\bar{O}_{n_{k+1}} \subset O_{n_k}$, $z \in \cap_k \bar{O}_{n_k} = \{\zeta\} \in \bar{W} \setminus \partial W$, which is a contradiction. Therefore, $O_n \subset W$ for $n$ large enough and $\partial O_n \cap F = \emptyset$ by construction, so we may set $O = O_n$ and $C := \partial O_n$ for any $n$ large enough, because $\partial O_n$ is a connected component of the level set $h^{-1}(c_n)$, and thus it is a smooth Jordan curve. This proves the claim.

If $\zeta \notin \text{rp}(\mathbb{R})$, let us pick $W$ so small that $p$ is injective on each connected component of $p^{-1}(p(W))$. We now argue as before, observing that $C \subset G_{\max}$ and $(p^{-1}(p(C)) \setminus C) \cap G_{\max} = \emptyset$ (by schlichtness), so that $G_{\max} \cap (p^{-1}(p(O)) \setminus O) = \emptyset$ because $G_{\max}$ is connected and $(p^{-1}(p(C)) \setminus C)$ is the boundary
Let $G_J$ be as in Lemma 3.12 for some small $\epsilon > 0$. Denote by $G_J^*$ the union of $G_J$ and the annular region delimited by $J$ and $\mathcal{T}$; the latter is diffeomorphic under $p$ to $\{z: 1 - \epsilon < |z| < 1\}$ if we fix $\epsilon$ small enough. Clearly, $G_J^*$ is a fine domain that lies schlicht over $\mathbb{D}$. Let $\mathcal{G}$ be the collection of all fine domains $G \subset \mathcal{R}$ lying schlicht over $\mathbb{D}$ and containing $G_J^*$. The set $\mathcal{G}$ is partially ordered by inclusion and every chain in it is bounded above by the union of its elements. Therefore, by Zorn's lemma, $\mathcal{G}$ possesses a maximal element, say $G_{\max}$. Note that if $\mathcal{R} \setminus G_{\max}$ is thin at two points $\zeta_1, \zeta_2 \in \mathcal{R}$, then $p(\zeta_1) \neq p(\zeta_2)$ as otherwise $G_{\max}$ could not be schlicht over $\mathbb{D}$ by Lemma A.2. Hence, it follows from (A.9) and the maximality of $G_{\max}$ that $\mathcal{R} \setminus G_{\max}$ is its own base.

**Lemma 3.14.** Any maximal domain $G_{\max} \in \mathcal{G}$ is a Euclidean domain. Moreover, no connected component of $\mathcal{R} \setminus G_{\max}$ (resp. $\mathbb{D} \setminus p(G_{\max})$) consists of a single point.

*Proof.* Observe that $G_{\max}$ cannot contain a ramification point of $\mathcal{R}$ as it is finely open and lies schlicht over $\mathbb{D}$, see Lemma A.4. Thus, for every $\zeta \in G_{\max}$ there exists $r_0 > 0$ such that each component of

$$p^{-1}(\{z \in \mathbb{D}: |z - p(\zeta)| < r_0\})$$

is in one-to-one correspondence with $\{z \in \mathbb{D}: |z - p(\zeta)| < r_0\}$ under $p$. Let $V$ be the component containing $\zeta$. Since the intersection $V \cap G_{\max}$ is finely open, $V \setminus G_{\max}$ is thin at $\zeta$ and therefore there is $r_1 \in (0, r_0)$ such that

$$V \cap p^{-1}(\{z \in \mathbb{D}: |z - p(\zeta)| = r_1\}) \subset G_{\max},$$

see Section A.6. Now, as $G_{\max}$ lies schlicht over $\mathbb{D}$, it holds that

$$(3.70) \qquad G_{\max} \cap (p^{-1}(\{z \in \mathbb{D}: |z - p(\zeta)| = r_1\}) \setminus V) = \emptyset.$$

Further, since $G_{\max}$ is finely connected, we necessarily have that

$$G_{\max} \cap (p^{-1}(\{z \in \mathbb{D}: |z - p(\zeta)| < r_1\}) \setminus V) = \emptyset,$$

otherwise $G_{\max}$ would intersect the fine boundary of $p^{-1}(\{z \in \mathbb{D}: |z - p(\zeta)| < r_1\}) \setminus V$ which is contained in $p^{-1}(\{z \in \mathbb{D}: |z - p(\zeta)| = r_1\}) \setminus V$ (in fact, equal to it by regularity), thereby contradicting (3.70). The maximality of $G_{\max}$ now yields that

$$p^{-1}(\{z \in \mathbb{D}: |z - \zeta| < r_1\}) \cap V \subset G_{\max},$$

hence $\zeta$ belongs to the Euclidean interior of $G_{\max}$ and so $G_{\max}$ is Euclidean open. Thus, its connected components are open and therefore finely open. Hence, $G_{\max}$ is a Euclidean domain since it is finely connected by definition.

To prove the second assertion, assume to the contrary that a point $\zeta \in \mathcal{R} \setminus G_{\max}$ is a connected component of the latter. We claim that any open neighborhood $W$ of $\zeta$ contains an open set $O \ni \zeta$ whose boundary $\partial O$ is a smooth Jordan curve $C$ contained in $G_{\max}$. To see this, assume with no loss of generality that $\bar{W} \subset \mathcal{R}$, and put $F := \mathcal{R} \setminus G_{\max} \cap \bar{W}$. The latter is closed in $\mathcal{R}$, so there is a smooth function $h: \mathcal{R} \to \mathbb{R}^+$ of which $F$ is the zero set, see Section 3.4. Given a sequence $\{c_n\}$ of regular values of $h$ tending to 0, let $O_n$ be the connected component containing $\zeta$ of the open set $\{z: h(z) < c_n\}$. The sets $O_n$ form a (strictly) nested sequence of connected open sets containing $\zeta$, whose intersection is connected and contained in $F$ whence reduces to $\zeta$. If $O_{n_k} \cap \partial W \neq \emptyset$ for some increasing subsequence of indices $n_k$, take $z_{n_k} \in O_{n_k} \cap \partial W$ and extract from $\{z_{n_k}\}$ a subsequence converging to $z \in \partial W$, which is possible due to compactness of the latter. On the other hand, since $\bar{O}_{n_{k+1}} \subset O_{n_k}$, $z \in \cap_k \bar{O}_{n_k} = \{\zeta\} \in \bar{W} \setminus \partial W$, which is a contradiction. Therefore, $O_n \subset W$ for $n$ large enough and $\partial O_n \cap F = \emptyset$ by construction, so we may set $O = O_n$ and $C := \partial O_n$ for any $n$ large enough, because $\partial O_n$ is a connected component of the level set $h^{-1}(c_n)$, and thus it is a smooth Jordan curve. This proves the claim.

If $\zeta \notin \text{rp}(\mathcal{R})$, let us pick $W$ so small that $p$ is injective on each connected component of $p^{-1}(p(W))$. We now argue as before, observing that $C \subset G_{\max}$ and $(p^{-1}(p(C)) \setminus C) \cap G_{\max} = \emptyset$ (by schlichtness), so that $G_{\max} \cap (p^{-1}(p(O)) \setminus O) = \emptyset$ because $G_{\max}$ is connected and $(p^{-1}(p(C)) \setminus C)$ is the boundary
of $p^{-1}(p(O))\backslash O$. Thus, by maximality, $G_{\max}$ should contain $O$ which contradicts the fact that $\zeta \in O$. Finally, if $\zeta \in \mathbf{rp}(\mathcal{R})$ then Lemma A.3 contradicts the existence of $C$.

This proves that no connected component of $\mathcal{R}\backslash G_{\max}$ consists of a single point. To show the same is true of $\mathcal{D}\backslash p(G_{\max})$, observe that such a component would consist, by the same reasoning as before, of a $z \in \partial p(G_{\max})$ lying interior to Jordan curves of arbitrary small diameter contained in $p(G_{\max})$. For $\gamma: [0, 2\pi] \to p(G_{\max})$ such a parametrized Jordan curve, let $\xi \in G_{\max}$ satisfy $p(\xi) = \gamma(0) = \gamma(2\pi)$. Let further $\ell: [0, 2\pi] \to \mathcal{R}$ be a continuous lift of $\gamma$ starting at $\xi$, i.e., $\ell(0) = \xi$ and $p(\ell(t)) = \gamma(t)$ for $t \in [0, 2\pi]$. Of necessity $\ell([0, 2\pi]) \subset G_{\max}$, because $p(\partial G_{\max}) \cap p(G_{\max}) = \emptyset$ since $G_{\max}$ is open and lies schlicht over $\mathcal{D}$, while $p$ is an open map; then schlichtness again implies that $\ell$ is a parametrized Jordan curve in $G_{\max}$, and it is the unique lift of $\gamma$ to $G_{\max}$.

Set $p^{-1}(z) = \{\zeta_1, \dots, \zeta_\ell\}$, and let $D_z$ be an open disk centered at $z$ such that each connected component of $p^{-1}(D_z)$ is (isomorphic to) a $m(\zeta_j)$-sheeted cyclic covering $D_{\zeta_j}$ of $D_z$; in addition, we require that $D_z$ is so small that $D_{\zeta_j} \subset U_{\zeta_j}$ for all $j$, where $U_{\zeta_j}$ is as in Lemma A.3. Let $\gamma: [0, 2\pi] \to D_z$ be a parametrized Jordan curve containing $z$ in its interior, and $\ell: [0, 2\pi] \to G_{\max}$ the associated lift. Necessarily $\ell([0, 2\pi])$ is contained in a single component of $p^{-1}(D_z)$, say $D_{\zeta_j}$, and it must contain $\zeta_j$ in its interior (otherwise $\ell$ would be a unit in $\pi_1(D_{\zeta_j} \setminus \zeta_j)$ and so would be $\gamma = p \circ \ell$ in $\pi_1(D_z \setminus z)$). Now, if $m(\zeta_j) > 1$, then we contradict Lemma A.3. Hence, $\ell$ is valued in a $D_{\zeta_j}$ such that $m(\zeta_j) = 1$, which is thus homeomorphic to $D_z$ under $p$.

Let $\gamma_n$ be a sequence of Jordan curves in $D_z$, containing $z$ in their interior and shrinking to $z$ when $n \to \infty$. Let further $\ell_n$ be the corresponding sequence of lifts to $G_{\max}$. By what precedes, some subsequence $\ell_{m_n}$ shrinks to $\zeta_j$ in $D_{\zeta_j}$, for some $j$ such that $D_{\zeta_j}$ is homeomorphic to $D_z$ under $p$. Moreover, $\ell_{m_n}$ contains $\zeta_j$ in its interior. We can now argue as we did to show that no connected component of $\mathcal{R}\backslash G_{\max}$ consists of a single point, and contradict the maximal character of $G_{\max}$. This completes the proof of the lemma. $\square$

Define $D_{\max} := p(G_{\max})$ and $V_{\max} := p^{-1}(D_{\max})$. Notice that both sets are open by Lemma 3.14 along with openness and continuity of $p$.

**Lemma 3.15.** Let $N_2 := p^{-1}(\tilde{N}_2)$, where $\tilde{N}_2$ is the $+\infty$-set of $g(\mu^{(2)}, \mathcal{D}; \cdot)$ and we set

$$ (3.71) \qquad \mu^{(2)} := (\mu^{(1)})^{\mathcal{D}\backslash D_{\max}} \quad \text{and} \quad \nu^{(2)} := (\nu^{(1)})^{\mathcal{R}\backslash V_{\max}}. $$

Then $N_2 = N_1\backslash V_{\max}$, and for $z \in \mathcal{R}\backslash N_2$ we can define

$$ (3.72) \qquad \mathrm{ler}^{(2)}(z) := g(\mu^{(2)}, \mathcal{D}; p(z)) - g(\nu^{(2)}, \mathcal{R}; z) - h_{\mathcal{R}}(z). $$

In this case it holds that

$$ (3.73) \qquad \left\{ \begin{array}{ll} \limsup_{\zeta \to z} \mathrm{ler}^{(2)}(\zeta) \le -2/\mathrm{cap}_{\mathcal{D}}(K_f), & z \in \mathcal{T}, \\ \mathrm{ler}^{(2)}(\zeta) = 0, & z \in \mathcal{R}\backslash(G_{\max} \cup N_2). \end{array} \right. $$

Moreover, we have that

$$ (3.74) \qquad \|\mu^{(2)}\| \begin{cases} = \|\mu\| & \text{if } \mathrm{supp}(\mu^{(1)}) \cap D_{\max} = \emptyset, \\ < \|\mu\| & \text{otherwise.} \end{cases} $$

*Proof.* By Lemma 3.14 the set $\mathcal{D}\backslash D_{\max}$ is closed in $\mathcal{D}$ and none of its connected components reduces to a point. Hence, it has no finely isolated points (remember that a connected set cannot be thin at an accumulation point, see discussion after (A.8)). Thus, $\mathcal{D}\backslash D_{\max}$ is its own base. Consequently, $\mathcal{R}\backslash V_{\max}$ is also its own base by Lemma A.2. Therefore, we get from (A.22) that

$$ (3.75) \qquad \mu^{(2)}(D_{\max}) = \nu^{(2)}(V_{\max}) = 0. $$

This implies that the potentials of $\mu^{(2)}$ and $\nu^{(2)}$ are harmonic in $D_{\max}$ and $V_{\max}$, respectively. Moreover, since $\partial D_{\max}\backslash \mathcal{T}$ is separated from $\mathcal{T}$ by the very definition of $G_{\max}$, $\partial V_{\max}\backslash \partial \mathcal{R}$ is necessarily
of $p^{-1}(p(O))\setminus O$. Thus, by maximality, $G_{\max}$ should contain $O$ which contradicts the fact that $\zeta \in O$. Finally, if $\zeta \in \mathbf{rp}(\mathcal{R})$ then Lemma A.3 contradicts the existence of $C$.

This proves that no connected component of $\mathcal{R}\setminus G_{\max}$ consists of a single point. To show the same is true of $\mathcal{D}\setminus p(G_{\max})$, observe that such a component would consist, by the same reasoning as before, of a $z \in \partial p(G_{\max})$ lying interior to Jordan curves of arbitrary small diameter contained in $p(G_{\max})$. For $\gamma: [0, 2\pi] \to p(G_{\max})$ such a parametrized Jordan curve, let $\xi \in G_{\max}$ satisfy $p(\xi) = \gamma(0) = \gamma(2\pi)$. Let further $\ell: [0, 2\pi] \to \mathcal{R}$ be a continuous lift of $\gamma$ starting at $\xi$, i.e., $\ell(0) = \xi$ and $p(\ell(t)) = \gamma(t)$ for $t \in [0, 2\pi]$. Of necessity $\ell([0, 2\pi]) \subset G_{\max}$, because $p(\partial G_{\max}) \cap p(G_{\max}) = \emptyset$ since $G_{\max}$ is open and lies schlicht over $\mathcal{D}$, while $p$ is an open map; then schlichtness again implies that $\ell$ is a parametrized Jordan curve in $G_{\max}$, and it is the unique lift of $\gamma$ to $G_{\max}$.

Set $p^{-1}(z) = \{\zeta_1, \dots, \zeta_\ell\}$, and let $D_z$ be an open disk centered at $z$ such that each connected component of $p^{-1}(D_z)$ is (isomorphic to) a $m(\zeta_j)$-sheeted cyclic covering $D_{\zeta_j}$ of $D_z$; in addition, we require that $D_z$ is so small that $D_{\zeta_j} \subset U_{\zeta_j}$ for all $j$, where $U_{\zeta_j}$ is as in Lemma A.3. Let $\gamma: [0, 2\pi] \to D_z$ be a parametrized Jordan curve containing $z$ in its interior, and $\ell: [0, 2\pi] \to G_{\max}$ the associated lift. Necessarily $\ell([0, 2\pi])$ is contained in a single component of $p^{-1}(D_z)$, say $D_{\zeta_j}$, and it must contain $\zeta_j$ in its interior (otherwise $\ell$ would be a unit in $\pi_1(D_{\zeta_j} \setminus \zeta_j)$ and so would be $\gamma = p \circ \ell$ in $\pi_1(D_z \setminus z)$). Now, if $m(\zeta_j) > 1$, then we contradict Lemma A.3. Hence, $\ell$ is valued in a $D_{\zeta_j}$ such that $m(\zeta_j) = 1$, which is thus homeomorphic to $D_z$ under $p$.

Let $\gamma_n$ be a sequence of Jordan curves in $D_z$, containing $z$ in their interior and shrinking to $z$ when $n \to \infty$. Let further $\ell_n$ be the corresponding sequence of lifts to $G_{\max}$. By what precedes, some subsequence $\ell_{m_n}$ shrinks to $\zeta_j$ in $D_{\zeta_j}$, for some $j$ such that $D_{\zeta_j}$ is homeomorphic to $D_z$ under $p$. Moreover, $\ell_{m_n}$ contains $\zeta_j$ in its interior. We can now argue as we did to show that no connected component of $\mathcal{R}\setminus G_{\max}$ consists of a single point, and contradict the maximal character of $G_{\max}$. This completes the proof of the lemma. $\square$

Define $D_{\max} := p(G_{\max})$ and $V_{\max} := p^{-1}(D_{\max})$. Notice that both sets are open by Lemma 3.14 along with openness and continuity of $p$.

**Lemma 3.15.** Let $N_2 := p^{-1}(\tilde{N}_2)$, where $\tilde{N}_2$ is the $+\infty$-set of $g(\mu^{(2)}, \mathcal{D}; \cdot)$ and we set

$$ (3.71) \qquad \mu^{(2)} := (\mu^{(1)})^{\mathcal{D}\setminus D_{\max}} \quad \text{and} \quad \nu^{(2)} := (\nu^{(1)})^{\mathcal{R}\setminus V_{\max}}. $$

Then $N_2 = N_1 \setminus V_{\max}$, and for $z \in \mathcal{R}\setminus N_2$ we can define

$$ (3.72) \qquad \mathrm{ler}^{(2)}(z) := g(\mu^{(2)}, \mathcal{D}; p(z)) - g(\nu^{(2)}, \mathcal{R}; z) - h_{\mathcal{R}}(z). $$

In this case it holds that

$$ (3.73) \qquad \left\{ \begin{array}{ll} \limsup_{\zeta \to z} \mathrm{ler}^{(2)}(\zeta) \le -2/\mathrm{cap}_{\mathcal{D}}(K_f), & z \in \mathcal{T}, \\ \mathrm{ler}^{(2)}(\zeta) = 0, & z \in \mathcal{R}\setminus(G_{\max} \cup N_2). \end{array} \right. $$

Moreover, we have that

$$ (3.74) \qquad \|\mu^{(2)}\| \begin{cases} = \|\mu\| & \text{if } \mathrm{supp}(\mu^{(1)}) \cap D_{\max} = \emptyset, \\ < \|\mu\| & \text{otherwise.} \end{cases} $$

*Proof.* By Lemma 3.14 the set $\mathcal{D}\setminus D_{\max}$ is closed in $\mathcal{D}$ and none of its connected components reduces to a point. Hence, it has no finely isolated points (remember that a connected set cannot be thin at an accumulation point, see discussion after (A.8)). Thus, $\mathcal{D}\setminus D_{\max}$ is its own base. Consequently, $\mathcal{R}\setminus V_{\max}$ is also its own base by Lemma A.2. Therefore, we get from (A.22) that

$$ (3.75) \qquad \mu^{(2)}(D_{\max}) = \nu^{(2)}(V_{\max}) = 0. $$

This implies that the potentials of $\mu^{(2)}$ and $\nu^{(2)}$ are harmonic in $D_{\max}$ and $V_{\max}$, respectively. Moreover, since $\partial D_{\max}\setminus \mathcal{T}$ is separated from $\mathcal{T}$ by the very definition of $G_{\max}$, $\partial V_{\max}\setminus \partial \mathcal{R}$ is necessarily
separated from $\partial\mathcal{R}$ and these potentials extend continuously by zero to $\mathbb{T}$ and $\partial\mathcal{R}$ respectively by the regularity of the latter, see the discussion after (A.12). Therefore, $ler^{(2)}$ extends harmonically to the whole of $V_{\max}$ and continuously to $\partial\mathcal{R}\setminus\mathcal{T}$ by Lemma 3.8. So, as in Lemma 3.13, the inclusion $N_2 \subseteq N_1\setminus V_{\max}$ follows directly from (A.21) and the equality $N_2 = N_1\setminus V_{\max}$ is then deduced from (A.22). In addition, the first inequality in (3.73) holds in view of (3.17), the definition of $h_{\mathcal{R}}$ given after (3.46), and the fact that $h''$ is non-negative, see Lemma 3.5.

Since $N_1\setminus V' = N_0\setminus V'$ by Lemma 3.13 we have that $\mathcal{R}\setminus(V' \cup V_{\max} \cup N_0) = \mathcal{R}\setminus(V' \cup V_{\max} \cup N_1)$. Moreover, $ler(z) = 0$ on this set by the very definitions of $V'$ and $V_{\max}$. Further,

$$ \mathcal{R}\setminus(V_{\max} \cup N_2) = \mathcal{R}\setminus(V_{\max} \cup N_1) = (\mathcal{R}\setminus(V' \cup V_{\max} \cup N_1)) \cup (V'\setminus N_1). $$

Since $\mathbb{D}\setminus D_{\max}$ and $\mathcal{R}\setminus V_{\max}$ are their own bases, it therefore follows from (A.22) and (3.68) that

$$ (3.76) \qquad ler^{(2)}(z) = ler^{(1)}(z) = 0, \quad z \in \mathcal{R}\setminus(V_{\max} \cup N_2). $$

To study the values of $ler^{(2)}$ on $V_{\max}\setminus G_{\max}$, let us show that this is an open set. Indeed, since $p(G_{\max}) = p(V_{\max}) = D_{\max}$, for each $\zeta \in V_{\max}$ there exists a disk $D_{p(\zeta)} \subseteq D_{\max}$, centered at $p(\zeta)$, and a point $z \in G_{\max}$ with $p(z) = p(\zeta)$ such that $D_{\zeta} \subseteq V_{\max}$ and $D_z \subseteq G_{\max}$, where $D_{\zeta}$ and $D_z$ are the connected components of $p^{-1}(D_{p(\zeta)})$ that contain $\zeta$ and $z$, respectively. If $\zeta \neq z$, then $D_{\zeta} \cap G_{\max} = \emptyset$ since $G_{\max}$ lies schlicht over $\mathbb{D}$ and therefore $D_z \subseteq V_{\max}\setminus G_{\max}$ as claimed. Moreover, since $\mathcal{R}\setminus V_{\max}$ is its own base, we get from Lemma A.1 that $\mathcal{R}\setminus(V_{\max}\setminus G_{\max})$ is also its own base.

Pick $z \in V_{\max}\setminus G_{\max}$ and notice that $V_{\max}\setminus G_{\max}$ consists of at most finitely many connected components. Let $V_z$ be the component containing $z$. As $G_{\max}$ contains the annular region delimited by $J$ and $\mathcal{T}$ and lies schlicht over $\mathbb{D}$, it follows that $\partial(V_{\max}\setminus G_{\max}) \cap \partial\mathcal{R} = \partial\mathcal{R}\setminus\mathcal{T}$. Since $h_{\mathcal{R}} = 0$ on $\partial\mathcal{R}\setminus\mathcal{T}$ by Lemma 3.8, we get from Lemmas A.6 and A.10 that

$$ (3.77) \qquad \int h_{\mathcal{R}}(x)d\delta_z^{\mathcal{R}\setminus V_z}(x) = \int h_{\mathcal{R}}(x)d\delta_z^{\mathcal{R}\setminus(V_{\max}\setminus G_{\max})}(x) = h_{\mathcal{R}}(z). $$

Since $V_z$, $V_{\max}\setminus G_{\max}$, and $G_{\max}$ are Euclidean open sets, it holds that $\partial V_z \subseteq \partial(V_{\max}\setminus G_{\max}) \subseteq \partial V_{\max}$, a set on which $ler^{(1)}(z)$ is zero except possibly on $N_2$ by (3.76). Consequently, in view of (3.72), we see upon using Lemma A.7, (A.25), and (3.77) that

$$ ler^{(2)}(z) = \int ler^{(1)}(x)d\delta_z^{\mathcal{R}\setminus V_z}(x) = 0, $$

where the second equality follows from Lemma A.6 as $\partial_t V_z \subseteq \partial V_z$ and $ler^{(1)}(x) = 0$ quasi and therefore $\delta_z^{\mathcal{R}\setminus V_z}$-almost everywhere on $\partial_t V_z$.

Finally, we get from (A.24) and (A.38) that $\|\mu^{(2)}\| = \|\mu^{(1)}\|$ when $\mathrm{supp}(\mu^{(1)}) \cap D_{\max} = \emptyset$ and $\|\mu^{(2)}\| < \|\mu^{(1)}\|$ otherwise as well as that $\|\mu^{(1)}\| = \|\mu\|$, which proves (3.74). $\square$

### 3.9. Projected Logarithmic Error Function.
Recall that $z \in \partial G_{\max}$ is called accessible if there exists a continuous map $\psi$ on $[0, 1]$ such that $\psi(t) \in G_{\max}$ for $t \in [0, 1)$ and $\psi(1) = z$.

**Lemma 3.16.** It holds that $\mathrm{card}(p^{-1}(z) \cap (\partial G_{\max}\setminus E^*)) \le 2$ for all except countably many $z \in \partial D_{\max}$, where $E^* \subseteq \partial G_{\max}$ is the subset of non-accessible points.

*Proof.* Since $\mathbb{T} \subset \partial D_{\max}$ and $p^{-1}(\mathbb{T}) \cap \partial G_{\max} = \mathcal{T}$, we only need to consider $z \in \partial D_{\max} \cap \mathbb{D}$. The set $(\partial D_{\max} \cap \mathbb{D})\setminus p(\mathbf{rp}(\mathcal{R}))$ can be covered by countably many open sets of the form $D_x$, where $x \in (\partial D_{\max} \cap \mathbb{D})\setminus p(\mathbf{rp}(\mathcal{R}))$ and $D_x \subset \mathbb{D}$ is a disk centered at $x$, small enough that each component of $p^{-1}(D_x)$ is homeomorphic to $D_x$ under $p$. Hence, it is enough to show that for each such $D_x$ and all $z \in \partial D_{\max} \cap D_x$ but countably many, one has card $(p^{-1}(z) \cap (\partial G_{\max}\setminus E^*)) \le 2$. Fix $D_x$ and let $x_1, \dots, x_N$ denote the preimages of $x$ under $p$. We write $V_{x_\ell}$ for the connected component of $p^{-1}(D_x)$ containing $x_\ell$, $1 \le \ell \le N$. Then, each $z \in D_x$ has preimages $z_1, \dots, z_N$ under $p$, with $z_\ell \in V_{x_\ell}$. When $z \in \partial D_{\max} \cap D_x$, it follows from the definition of accessibility that if $z_\ell \in \partial G_{\max}\setminus E^*$, then
separated from $\partial\mathcal{R}$ and these potentials extend continuously by zero to $\mathbb{T}$ and $\partial\mathcal{R}$ respectively by the regularity of the latter, see the discussion after (A.12). Therefore, $ler^{(2)}$ extends harmonically to the whole of $V_{\max}$ and continuously to $\partial\mathcal{R}\setminus\mathcal{T}$ by Lemma 3.8. So, as in Lemma 3.13, the inclusion $N_2 \subseteq N_1\setminus V_{\max}$ follows directly from (A.21) and the equality $N_2 = N_1\setminus V_{\max}$ is then deduced from (A.22). In addition, the first inequality in (3.73) holds in view of (3.17), the definition of $h_{\mathcal{R}}$ given after (3.46), and the fact that $h''$ is non-negative, see Lemma 3.5.

Since $N_1\setminus V' = N_0\setminus V'$ by Lemma 3.13 we have that $\mathcal{R}\setminus(V' \cup V_{\max} \cup N_0) = \mathcal{R}\setminus(V' \cup V_{\max} \cup N_1)$. Moreover, $ler(z) = 0$ on this set by the very definitions of $V'$ and $V_{\max}$. Further,

$$ \mathcal{R}\setminus(V_{\max} \cup N_2) = \mathcal{R}\setminus(V_{\max} \cup N_1) = (\mathcal{R}\setminus(V' \cup V_{\max} \cup N_1)) \cup (V'\setminus N_1). $$

Since $\mathbb{D}\setminus D_{\max}$ and $\mathcal{R}\setminus V_{\max}$ are their own bases, it therefore follows from (A.22) and (3.68) that

$$ (3.76) \qquad ler^{(2)}(z) = ler^{(1)}(z) = 0, \quad z \in \mathcal{R}\setminus(V_{\max} \cup N_2). $$

To study the values of $ler^{(2)}$ on $V_{\max}\setminus G_{\max}$, let us show that this is an open set. Indeed, since $p(G_{\max}) = p(V_{\max}) = D_{\max}$, for each $\zeta \in V_{\max}$ there exists a disk $D_{p(\zeta)} \subseteq D_{\max}$, centered at $p(\zeta)$, and a point $z \in G_{\max}$ with $p(z) = p(\zeta)$ such that $D_{\zeta} \subseteq V_{\max}$ and $D_z \subseteq G_{\max}$, where $D_{\zeta}$ and $D_z$ are the connected components of $p^{-1}(D_{p(\zeta)})$ that contain $\zeta$ and $z$, respectively. If $\zeta \neq z$, then $D_{\zeta} \cap G_{\max} = \emptyset$ since $G_{\max}$ lies schlicht over $\mathbb{D}$ and therefore $D_z \subseteq V_{\max}\setminus G_{\max}$ as claimed. Moreover, since $\mathcal{R}\setminus V_{\max}$ is its own base, we get from Lemma A.1 that $\mathcal{R}\setminus(V_{\max}\setminus G_{\max})$ is also its own base.

Pick $z \in V_{\max}\setminus G_{\max}$ and notice that $V_{\max}\setminus G_{\max}$ consists of at most finitely many connected components. Let $V_z$ be the component containing $z$. As $G_{\max}$ contains the annular region delimited by $J$ and $\mathcal{T}$ and lies schlicht over $\mathbb{D}$, it follows that $\partial(V_{\max}\setminus G_{\max}) \cap \partial\mathcal{R} = \partial\mathcal{R}\setminus\mathcal{T}$. Since $h_{\mathcal{R}} = 0$ on $\partial\mathcal{R}\setminus\mathcal{T}$ by Lemma 3.8, we get from Lemmas A.6 and A.10 that

$$ (3.77) \qquad \int h_{\mathcal{R}}(x)d\delta_z^{\mathcal{R}\setminus V_z}(x) = \int h_{\mathcal{R}}(x)d\delta_z^{\mathcal{R}\setminus(V_{\max}\setminus G_{\max})}(x) = h_{\mathcal{R}}(z). $$

Since $V_z$, $V_{\max}\setminus G_{\max}$, and $G_{\max}$ are Euclidean open sets, it holds that $\partial V_z \subseteq \partial(V_{\max}\setminus G_{\max}) \subseteq \partial V_{\max}$, a set on which $ler^{(1)}(z)$ is zero except possibly on $N_2$ by (3.76). Consequently, in view of (3.72), we see upon using Lemma A.7, (A.25), and (3.77) that

$$ ler^{(2)}(z) = \int ler^{(1)}(x)d\delta_z^{\mathcal{R}\setminus V_z}(x) = 0, $$

where the second equality follows from Lemma A.6 as $\partial_t V_z \subseteq \partial V_z$ and $ler^{(1)}(x) = 0$ quasi and therefore $\delta_z^{\mathcal{R}\setminus V_z}$-almost everywhere on $\partial_t V_z$.

Finally, we get from (A.24) and (A.38) that $\|\mu^{(2)}\| = \|\mu^{(1)}\|$ when $\mathrm{supp}(\mu^{(1)}) \cap D_{\max} = \emptyset$ and $\|\mu^{(2)}\| < \|\mu^{(1)}\|$ otherwise as well as that $\|\mu^{(1)}\| = \|\mu\|$, which proves (3.74). $\square$

### 3.9. Projected Logarithmic Error Function.
Recall that $z \in \partial G_{\max}$ is called accessible if there exists a continuous map $\psi$ on $[0, 1]$ such that $\psi(t) \in G_{\max}$ for $t \in [0, 1)$ and $\psi(1) = z$.

**Lemma 3.16.** It holds that $\mathrm{card}(p^{-1}(z) \cap (\partial G_{\max}\setminus E^*)) \le 2$ for all except countably many $z \in \partial D_{\max}$, where $E^* \subseteq \partial G_{\max}$ is the subset of non-accessible points.

*Proof.* Since $\mathbb{T} \subset \partial D_{\max}$ and $p^{-1}(\mathbb{T}) \cap \partial G_{\max} = \mathcal{T}$, we only need to consider $z \in \partial D_{\max} \cap \mathbb{D}$. The set $(\partial D_{\max} \cap \mathbb{D})\setminus p(\mathbf{rp}(\mathcal{R}))$ can be covered by countably many open sets of the form $D_x$, where $x \in (\partial D_{\max} \cap \mathbb{D})\setminus p(\mathbf{rp}(\mathcal{R}))$ and $D_x \subset \mathbb{D}$ is a disk centered at $x$, small enough that each component of $p^{-1}(D_x)$ is homeomorphic to $D_x$ under $p$. Hence, it is enough to show that for each such $D_x$ and all $z \in \partial D_{\max} \cap D_x$ but countably many, one has card $(p^{-1}(z) \cap (\partial G_{\max}\setminus E^*)) \le 2$. Fix $D_x$ and let $x_1, \dots, x_N$ denote the preimages of $x$ under $p$. We write $V_{x_\ell}$ for the connected component of $p^{-1}(D_x)$ containing $x_\ell$, $1 \le \ell \le N$. Then, each $z \in D_x$ has preimages $z_1, \dots, z_N$ under $p$, with $z_\ell \in V_{x_\ell}$. When $z \in \partial D_{\max} \cap D_x$, it follows from the definition of accessibility that if $z_\ell \in \partial G_{\max}\setminus E^*$, then
there is a continuous arc $\psi_{z\ell} : [0, 1] \to \bar{G}_{\max}$ such that $\psi_{z\ell}(t) \in G_{\max}$ for $t \in [0, 1)$ and $\psi_{z\ell}(1) = z\ell$. Moreover, on shortening and reparametrizing the arc if necessary, we may assume that $\psi_{z\ell}$ is valued in $V_{x\ell}$. Now, if $p^{-1}(z) \cap (\partial G_{\max} \setminus E^*)$ contains 3 distinct points, say $z_{\ell j}$ for $j = \{1, 2, 3\}$, then $p(\psi_{z_{\ell j}})$ are Jordan arcs having only the point $z$ in common, because $G_{\max}$ lies schlicht over $\mathbb{D}$ and the $V_{x\ell j}$ are disjoint. Thus, $T_z := \cup_j p(\psi_{z_{\ell j}})$ is a triod, and if we had uncountably many such $z$ then some triple $(\ell_1, \ell_2, \ell_3)$ would occur uncountably many times. Assigning different colors to $\psi_{z_{\ell_1}}, \psi_{z_{\ell_2}}$ and $\psi_{z_{\ell_3}}$, we get and uncountable collection of colored triods $T_z$ whose arcs of different colors never meet, again because $G_{\max}$ lies schlicht over $\mathbb{D}$ and the $V_{x\ell j}$ are disjoint. However, this contradicts the Moore triod theorem [50, Proposition 2.18], thereby finishing the proof. $\square$

Put $K_{\max} := \mathbb{D} \setminus D_{\max}$. It follows from Lemma 3.14 that $f$ has a single-valued meromorphic continuation throughout $D_{\max} \setminus p(E_f)$ and $K_{\max}$ is a compact subset of $\mathbb{D}$. Hence, $K_{\max} \in \mathcal{K}_f$, where $\mathcal{K}_f$ was defined just before (2.5). Clearly, the measure $\mu^{(2)}$ is supported on $K_{\max}$.

**Lemma 3.17.** Let $m_\zeta$ be the ramification order of $\zeta \in \mathcal{R}$. Define

$$ (3.78) \qquad \mathrm{ler}_{\mathrm{pr}}(z) := \sum_{\zeta \in p^{-1}(z)} m_{\zeta} \mathrm{ler}^{(2)}(\zeta), \quad z \in \mathbb{D} \setminus \tilde{N}_2. $$

Then $\mathrm{ler}_{\mathrm{pr}}$ is a $\delta$-subharmonic function in $\mathbb{D}$ such that $\mathrm{ler}_{\mathrm{pr}}(z) = 0$ when $z \in K_{\max} \setminus \tilde{N}_2$ and
$$ \limsup_{z \to \zeta} \mathrm{ler}_{\mathrm{pr}}(z) \le -2/\mathrm{cap}_{\mathbb{D}}(K_f), \quad \zeta \in \mathbb{T}. $$

Moreover, there exist non-negative measures $\mu_{\mathrm{pr}}$ and $v_{\mathrm{pr}}$, supported on $\partial D_{\max}$, and a non-negative function $h_{\mathrm{pr}}$, harmonic in $\mathbb{D}$, such that $\mu_{\mathrm{pr}} \le 2\mu^{(2)}$ and for $z \in \mathbb{D} \setminus \tilde{N}_2$

$$ (3.79) \qquad \mathrm{ler}_{\mathrm{pr}}(z) = g(\mu_{\mathrm{pr}}, \mathbb{D}; z) - g(v_{\mathrm{pr}}, \mathbb{D}; z) - h_{\mathrm{pr}}(z) - 2/\mathrm{cap}_{\mathbb{D}}(K_f). $$

*Proof.* The first two claims of the lemma follow readily from (3.73) and the computation in (A.11). Let $\hat{\mu}^{(2)}$ be the pullback of $\mu^{(2)}$ onto $\mathcal{R}$, see (A.33) and (A.34). Then we can equivalently write

$$ \begin{aligned} \mathrm{ler}^{(2)}(z) &= g(\hat{\mu}^{(2)}, \mathcal{R}; z) - g(v^{(2)}, \mathcal{R}; z) - h_{\mathcal{R}}(z) \\ &= g(\mu_{\mathcal{R}}, \mathcal{R}; z) - g(v_{\mathcal{R}}, \mathcal{R}; z) - h_{\mathcal{R}}(z), \end{aligned} $$

where $\mu_{\mathcal{R}} - v_{\mathcal{R}} = \hat{\mu}^{(2)} - v^{(2)}$ and $\mu_{\mathcal{R}}$, $v_{\mathcal{R}}$ are positive mutually singular measures on $p^{-1}(K_{\max})$, i.e., $\mu_{\mathcal{R}} - v_{\mathcal{R}}$ is the Riesz charge of a $\delta$-subharmonic function $\mathrm{ler}^{(2)}$. Clearly, $v_{\mathcal{R}} \le v^{(2)}$, $\mu_{\mathcal{R}} \le \hat{\mu}^{(2)}$, and for any Borel set $B \subset p^{-1}(K_{\max}) \setminus \mathbf{rp}(\mathcal{R})$ that lies schlicht over $\mathbb{D}$ it holds that

$$ (3.80) \qquad \mu_{\mathcal{R}}(B) \le \mu^{(2)}(p(B)). $$

Similarly to the computation in (A.11), one can show that $\sum_{\zeta \in p^{-1}(z)} m_\zeta h(\zeta)$ is harmonic in $\mathbb{D}$ when $h$ is harmonic on $\mathcal{R}$. Hence, it follows from the maximum principle for harmonic functions and Lemma 3.2 that

$$ \sum_{\zeta \in p^{-1}(z)} m_{\zeta} h_{\mathcal{R}}(\zeta) = 2/\mathrm{cap}_{\mathbb{D}}(K_f) + h_{\mathrm{pr}}(z), \quad h_{\mathrm{pr}}(z) := \sum_{\zeta \in p^{-1}(z)} m_{\zeta} h''(\zeta). $$

Thus, if we set $\mu_{\mathrm{pr}} = p_*(\mu_{\mathcal{R}})$ and $v_{\mathrm{pr}} = p_*(v_{\mathcal{R}})$, see (A.32), we get (3.79).

As explained in Lemma 3.15, $\mathcal{R} \setminus V_{\max}$ is its own base and $V_{\max} \setminus G_{\max}$, $G_{\max}$ are disjoint open sets. Hence, $\mathcal{R} \setminus G_{\max}$ is its own base by Lemma A.1. Moreover, the function $\mathrm{ler}^{(2)}$ is equal to zero on this set by (3.73). Hence, we get from the proof of [13, Theorem 2] (that carries over *mutatis mutandis* to any hyperbolic surface) the implication:

$$ (3.81) \qquad \delta_{z}^{\mathcal{R} \setminus G_{\max}}(F) = 0 \quad \Rightarrow \quad (\mu_{\mathcal{R}} + v_{\mathcal{R}})(F) = 0 $$

for any $F \subset \mathcal{R} \setminus G_{\max}$ and some (therefore any) $z \in G_{\max}$. In particular, it follows from Lemma A.6 that $\mathrm{supp}(\mu_{\mathcal{R}} + v_{\mathcal{R}}) \subseteq \partial G_{\max} \setminus \mathcal{T}$ and that this measure does not charge polar sets. Hence, since $G_{\max}$
there is a continuous arc $\psi_{z\ell} : [0, 1] \to \overline{G}_{\max}$ such that $\psi_{z\ell}(t) \in G_{\max}$ for $t \in [0, 1)$ and $\psi_{z\ell}(1) = z\ell$. Moreover, on shortening and reparametrizing the arc if necessary, we may assume that $\psi_{z\ell}$ is valued in $V_{x\ell}$. Now, if $p^{-1}(z) \cap (\partial G_{\max} \setminus E^*)$ contains 3 distinct points, say $z_{\ell j}$ for $j = \{1, 2, 3\}$, then $p(\psi_{z_{\ell j}})$ are Jordan arcs having only the point $z$ in common, because $G_{\max}$ lies schlicht over $\mathbb{D}$ and the $V_{x\ell j}$ are disjoint. Thus, $T_z := \cup_j p(\psi_{z_{\ell j}})$ is a triod, and if we had uncountably many such $z$ then some triple $(\ell_1, \ell_2, \ell_3)$ would occur uncountably many times. Assigning different colors to $\psi_{z_{\ell_1}}, \psi_{z_{\ell_2}}$ and $\psi_{z_{\ell_3}}$, we get and uncountable collection of colored triods $T_z$ whose arcs of different colors never meet, again because $G_{\max}$ lies schlicht over $\mathbb{D}$ and the $V_{x\ell j}$ are disjoint. However, this contradicts the Moore triod theorem [50, Proposition 2.18], thereby finishing the proof. $\square$

Put $K_{\max} := \mathbb{D} \setminus D_{\max}$. It follows from Lemma 3.14 that $f$ has a single-valued meromorphic continuation throughout $D_{\max} \setminus p(E_f)$ and $K_{\max}$ is a compact subset of $\mathbb{D}$. Hence, $K_{\max} \in \mathcal{K}_f$, where $\mathcal{K}_f$ was defined just before (2.5). Clearly, the measure $\mu^{(2)}$ is supported on $K_{\max}$.

**Lemma 3.17.** Let $m_\zeta$ be the ramification order of $\zeta \in \mathcal{R}$. Define

$$ (3.78) \qquad \mathrm{ler}_{\mathrm{pr}}(z) := \sum_{\zeta \in p^{-1}(z)} m_{\zeta} \mathrm{ler}^{(2)}(\zeta), \quad z \in \mathbb{D} \setminus \tilde{N}_2. $$

Then $\mathrm{ler}_{\mathrm{pr}}$ is a $\delta$-subharmonic function in $\mathbb{D}$ such that $\mathrm{ler}_{\mathrm{pr}}(z) = 0$ when $z \in K_{\max} \setminus \tilde{N}_2$ and
$$ \limsup_{z \to \zeta} \mathrm{ler}_{\mathrm{pr}}(z) \le -2/\mathrm{cap}_{\mathbb{D}}(K_f), \quad \zeta \in \mathbb{T}. $$

Moreover, there exist non-negative measures $\mu_{\mathrm{pr}}$ and $v_{\mathrm{pr}}$, supported on $\partial D_{\max}$, and a non-negative function $h_{\mathrm{pr}}$, harmonic in $\mathbb{D}$, such that $\mu_{\mathrm{pr}} \le 2\mu^{(2)}$ and for $z \in \mathbb{D} \setminus \tilde{N}_2$

$$ (3.79) \qquad \mathrm{ler}_{\mathrm{pr}}(z) = g(\mu_{\mathrm{pr}}, \mathbb{D}; z) - g(v_{\mathrm{pr}}, \mathbb{D}; z) - h_{\mathrm{pr}}(z) - 2/\mathrm{cap}_{\mathbb{D}}(K_f). $$

*Proof.* The first two claims of the lemma follow readily from (3.73) and the computation in (A.11). Let $\hat{\mu}^{(2)}$ be the pullback of $\mu^{(2)}$ onto $\mathcal{R}$, see (A.33) and (A.34). Then we can equivalently write

$$ \begin{aligned} \mathrm{ler}^{(2)}(z) &= g(\hat{\mu}^{(2)}, \mathcal{R}; z) - g(v^{(2)}, \mathcal{R}; z) - h_{\mathcal{R}}(z) \\ &= g(\mu_{\mathcal{R}}, \mathcal{R}; z) - g(v_{\mathcal{R}}, \mathcal{R}; z) - h_{\mathcal{R}}(z), \end{aligned} $$

where $\mu_{\mathcal{R}} - v_{\mathcal{R}} = \hat{\mu}^{(2)} - v^{(2)}$ and $\mu_{\mathcal{R}}$, $v_{\mathcal{R}}$ are positive mutually singular measures on $p^{-1}(K_{\max})$, i.e., $\mu_{\mathcal{R}} - v_{\mathcal{R}}$ is the Riesz charge of a $\delta$-subharmonic function $\mathrm{ler}^{(2)}$. Clearly, $v_{\mathcal{R}} \le v^{(2)}$, $\mu_{\mathcal{R}} \le \hat{\mu}^{(2)}$, and for any Borel set $B \subset p^{-1}(K_{\max}) \setminus \mathbf{rp}(\mathcal{R})$ that lies schlicht over $\mathbb{D}$ it holds that

$$ (3.80) \qquad \mu_{\mathcal{R}}(B) \le \mu^{(2)}(p(B)). $$

Similarly to the computation in (A.11), one can show that $\sum_{\zeta \in p^{-1}(z)} m_\zeta h(\zeta)$ is harmonic in $\mathbb{D}$ when $h$ is harmonic on $\mathcal{R}$. Hence, it follows from the maximum principle for harmonic functions and Lemma 3.2 that

$$ \sum_{\zeta \in p^{-1}(z)} m_{\zeta} h_{\mathcal{R}}(\zeta) = 2/\mathrm{cap}_{\mathbb{D}}(K_f) + h_{\mathrm{pr}}(z), \quad h_{\mathrm{pr}}(z) := \sum_{\zeta \in p^{-1}(z)} m_{\zeta} h''(\zeta). $$

Thus, if we set $\mu_{\mathrm{pr}} = p_*(\mu_{\mathcal{R}})$ and $v_{\mathrm{pr}} = p_*(v_{\mathcal{R}})$, see (A.32), we get (3.79).

As explained in Lemma 3.15, $\mathcal{R} \setminus V_{\max}$ is its own base and $V_{\max} \setminus G_{\max}$, $G_{\max}$ are disjoint open sets. Hence, $\mathcal{R} \setminus G_{\max}$ is its own base by Lemma A.1. Moreover, the function $\mathrm{ler}^{(2)}$ is equal to zero on this set by (3.73). Hence, we get from the proof of [13, Theorem 2] (that carries over *mutatis mutandis* to any hyperbolic surface) the implication:

$$ (3.81) \qquad \delta_{z}^{\mathcal{R} \setminus G_{\max}}(F) = 0 \quad \Rightarrow \quad (\mu_{\mathcal{R}} + v_{\mathcal{R}})(F) = 0 $$

for any $F \subset \mathcal{R} \setminus G_{\max}$ and some (therefore any) $z \in G_{\max}$. In particular, it follows from Lemma A.6 that $\mathrm{supp}(\mu_{\mathcal{R}} + v_{\mathcal{R}}) \subseteq \partial G_{\max} \setminus \mathcal{T}$ and that this measure does not charge polar sets. Hence, since $G_{\max}$
lies schlicht over $D_{\max}$, it holds that $\text{supp}(\mu_{\text{pr}} + v_{\text{pr}}) \subseteq \partial D_{\max} \setminus \mathbb{T}$. Moreover, let $\tilde{E}$ be the set of points $z \in \partial D_{\max}$ such that card $(p^{-1}(z) \cap (\partial G_{\max} \setminus E^*)) > 2$, where $E^*$ is the set of non-accessible points of $\partial G_{\max}$. It follows from Lemma 3.16 that $\tilde{E}$ is at most countable and therefore is not charged by $\mu_{\text{pr}}$. Of course, the same is true of $p(\mathbf{rp}(\mathcal{R}))$, as it is a finite set. For any Borel set $B \subseteq \partial D_{\max}$, let $\tilde{B} := B \setminus (\tilde{E} \cup p(\mathbf{rp}(\mathcal{R})))$. Then, we obtain

$$ 
\begin{aligned}
\mu_{\text{pr}}(B) &= \mu_{\text{pr}}(\tilde{B}) = \mu_{\mathcal{R}}(p^{-1}(\tilde{B}) \cap \partial G_{\max}) \\
&= \mu_{\mathcal{R}}(p^{-1}(\tilde{B}) \cap (\partial G_{\max} \setminus E^*)) \le 2\mu^{(2)}(B),
\end{aligned}
$$

where the third equality follows from [13, Corollary 2], which says that the Riesz charge of a $\delta$-subharmonic function cannot charge points that are non-accessible from the complement of the base of its zero set, and the last inequality is due to Lemma 3.16 and (3.80). $\square$

3.10. **Computation of the Logarithmic Error Function.** Lemma 3.17 implies that $\text{ler}_{\text{pr}}(z) = 0$ for $z \in K_{\max} \setminus \tilde{N}_2$. Hence, it follows from (3.79) and properties of the Green equilibrium potential that

$$ (3.82) \qquad g(\mu_{\text{pr}}, \mathbb{D}; z) - g(v_{\text{pr}}, \mathbb{D}; z) - h_{\text{pr}}(z) = 2 \frac{\text{cap}_{\mathbb{D}}(K_{\max})}{\text{cap}_{\mathbb{D}}(K_f)} g(\mu_{\mathbb{D}, K_{\max}}, \mathbb{D}; z) $$

for $z \in K_{\max} \setminus \tilde{N}_2$ (since $K_{\max}$ is equal to its own base, $g(\mu_{\mathbb{D}, K_{\max}}, \mathbb{D}; \cdot) \equiv 1/\text{cap}_{\mathbb{D}}(K_{\max})$ on $K_{\max}$). Assume that either $v_{\text{pr}}$ is a non-zero measure or $h_{\text{pr}}$ is strictly positive (harmonic) function in $\mathbb{D}$. Then, since $K_{\max}$ is separated from $\mathbb{T}$, it would hold that

$$ g(\mu_{\text{pr}}, \mathbb{D}; z) > 2 \frac{\text{cap}_{\mathbb{D}}(K_{\max})}{\text{cap}_{\mathbb{D}}(K_f)} g(\mu_{\mathbb{D}, K_{\max}}, \mathbb{D}; z) $$

for $z \in K_{\max} \setminus \tilde{N}_2$. Since $\text{supp}(\mu_{\mathbb{D}, K_{\max}}) \subseteq K_{\max}$ and $K_{\max}$ is its own base, the Strong Domination Principle, see Section A.6, implies that the above inequality holds everywhere in $\mathbb{D}$. Thus, integrating both sides of this inequality against $\mu_{\mathbb{D}, K_{\max}}$ which is a probability measure, we get on using Tonelli's theorem on the left-hand side and multiplying by $\text{cap}_{\mathbb{D}}(K_{\max})$ that

$$ (3.83) \qquad \|\mu_{\text{pr}}\| > 2 \frac{\text{cap}_{\mathbb{D}}(K_{\max})}{\text{cap}_{\mathbb{D}}(K_f)}. $$

On the one hand, by Lemma 3.17 and equation (3.74) together with the very construction of $\mu$, we have that $\|\mu_{\text{pr}}\| \le 2\|\mu^{(2)}\| \le 2\|\mu\| \le 2$. On the other hand holds $\text{cap}_{\mathbb{D}}(K_{\max}) \ge \text{cap}_{\mathbb{D}}(K_f)$, see (2.5). These observations clearly show that (3.83) is impossible. Hence, it is necessarily the case that

$$ (3.84) \qquad v_{\text{pr}} = 0 \quad \text{and} \quad h_{\text{pr}}(z) = 0, \quad z \in \mathbb{D}. $$

Then, one can rewrite (3.82) as

$$ (3.85) \qquad g(\mu_{\text{pr}}, \mathbb{D}; z) = 2 \frac{\text{cap}_{\mathbb{D}}(K_{\max})}{\text{cap}_{\mathbb{D}}(K_f)} g(\mu_{\mathbb{D}, K_{\max}}, \mathbb{D}; z) $$

for $z \in K_{\max} \setminus \tilde{N}_2$. Using now the Strong Domination Principle in both directions, we get that (3.85) holds for every $z \in \mathbb{D}$. Therefore,

$$ (3.86) \qquad \mu^{(2)} \ge \frac{1}{2} \mu_{\text{pr}} = \frac{\text{cap}_{\mathbb{D}}(K_{\max})}{\text{cap}_{\mathbb{D}}(K_f)} \mu_{\mathbb{D}, K_{\max}} $$

which, upon recalling once again (2.5) and the fact that $\mu_{\mathbb{D}, K_{\max}}$ is a probability measure, gives us

$$ (3.87) \qquad \text{cap}_{\mathbb{D}}(K_{\max}) = \text{cap}_{\mathbb{D}}(K_f) \quad \text{and} \quad \mu^{(2)} = \mu_{\mathbb{D}, K_{\max}}. $$

In particular $\tilde{N}_2 = \emptyset$ and the first equality in (3.87), combined with the minimality and uniqueness of $K_f$, yields that $K_{\max} = K_f$. In addition, as $\|\mu^{(2)}\| = 1$ by (3.87), we get from (3.74) that $\text{supp}(\mu^{(1)}) \cap D_{\max} = \emptyset$ and therefore $\mu^{(1)} = \mu^{(2)}$, by (A.22), (3.71), and the fact that $\mathbb{D} \setminus D_{\max}$ is its own base (because so is $\mathcal{R} \setminus G_{\max}$ and we can use Lemma A.2). Moreover, we also get from (3.74) that
lies schlicht over $D_{\max}$, it holds that $\text{supp}(\mu_{\text{pr}} + v_{\text{pr}}) \subseteq \partial D_{\max} \setminus \mathbb{T}$. Moreover, let $\tilde{E}$ be the set of points $z \in \partial D_{\max}$ such that card $(p^{-1}(z) \cap (\partial G_{\max} \setminus E^*)) > 2$, where $E^*$ is the set of non-accessible points of $\partial G_{\max}$. It follows from Lemma 3.16 that $\tilde{E}$ is at most countable and therefore is not charged by $\mu_{\text{pr}}$. Of course, the same is true of $p(\mathbf{rp}(\mathcal{R}))$, as it is a finite set. For any Borel set $B \subseteq \partial D_{\max}$, let $\tilde{B} := B \setminus (\tilde{E} \cup p(\mathbf{rp}(\mathcal{R})))$. Then, we obtain

$$ 
\begin{aligned}
\mu_{\text{pr}}(B) &= \mu_{\text{pr}}(\tilde{B}) = \mu_{\mathcal{R}}(p^{-1}(\tilde{B}) \cap \partial G_{\max}) \\
&= \mu_{\mathcal{R}}(p^{-1}(\tilde{B}) \cap (\partial G_{\max} \setminus E^*)) \le 2\mu^{(2)}(B),
\end{aligned}
$$

where the third equality follows from [13, Corollary 2], which says that the Riesz charge of a $\delta$-subharmonic function cannot charge points that are non-accessible from the complement of the base of its zero set, and the last inequality is due to Lemma 3.16 and (3.80). $\square$

3.10. **Computation of the Logarithmic Error Function.** Lemma 3.17 implies that $\text{ler}_{\text{pr}}(z) = 0$ for $z \in K_{\max} \setminus \tilde{N}_2$. Hence, it follows from (3.79) and properties of the Green equilibrium potential that

$$ (3.82) \qquad g(\mu_{\text{pr}}, \mathbb{D}; z) - g(v_{\text{pr}}, \mathbb{D}; z) - h_{\text{pr}}(z) = 2 \frac{\text{cap}_{\mathbb{D}}(K_{\max})}{\text{cap}_{\mathbb{D}}(K_f)} g(\mu_{\mathbb{D}, K_{\max}}, \mathbb{D}; z) $$

for $z \in K_{\max} \setminus \tilde{N}_2$ (since $K_{\max}$ is equal to its own base, $g(\mu_{\mathbb{D}, K_{\max}}, \mathbb{D}; \cdot) \equiv 1/\text{cap}_{\mathbb{D}}(K_{\max})$ on $K_{\max}$). Assume that either $v_{\text{pr}}$ is a non-zero measure or $h_{\text{pr}}$ is strictly positive (harmonic) function in $\mathbb{D}$. Then, since $K_{\max}$ is separated from $\mathbb{T}$, it would hold that

$$ g(\mu_{\text{pr}}, \mathbb{D}; z) > 2 \frac{\text{cap}_{\mathbb{D}}(K_{\max})}{\text{cap}_{\mathbb{D}}(K_f)} g(\mu_{\mathbb{D}, K_{\max}}, \mathbb{D}; z) $$

for $z \in K_{\max} \setminus \tilde{N}_2$. Since $\text{supp}(\mu_{\mathbb{D}, K_{\max}}) \subseteq K_{\max}$ and $K_{\max}$ is its own base, the Strong Domination Principle, see Section A.6, implies that the above inequality holds everywhere in $\mathbb{D}$. Thus, integrating both sides of this inequality against $\mu_{\mathbb{D}, K_{\max}}$ which is a probability measure, we get on using Tonelli's theorem on the left-hand side and multiplying by $\text{cap}_{\mathbb{D}}(K_{\max})$ that

$$ (3.83) \qquad \|\mu_{\text{pr}}\| > 2 \frac{\text{cap}_{\mathbb{D}}(K_{\max})}{\text{cap}_{\mathbb{D}}(K_f)}. $$

On the one hand, by Lemma 3.17 and equation (3.74) together with the very construction of $\mu$, we have that $\|\mu_{\text{pr}}\| \le 2\|\mu^{(2)}\| \le 2\|\mu\| \le 2$. On the other hand holds $\text{cap}_{\mathbb{D}}(K_{\max}) \ge \text{cap}_{\mathbb{D}}(K_f)$, see (2.5). These observations clearly show that (3.83) is impossible. Hence, it is necessarily the case that

$$ (3.84) \qquad v_{\text{pr}} = 0 \quad \text{and} \quad h_{\text{pr}}(z) = 0, \quad z \in \mathbb{D}. $$

Then, one can rewrite (3.82) as

$$ (3.85) \qquad g(\mu_{\text{pr}}, \mathbb{D}; z) = 2 \frac{\text{cap}_{\mathbb{D}}(K_{\max})}{\text{cap}_{\mathbb{D}}(K_f)} g(\mu_{\mathbb{D}, K_{\max}}, \mathbb{D}; z) $$

for $z \in K_{\max} \setminus \tilde{N}_2$. Using now the Strong Domination Principle in both directions, we get that (3.85) holds for every $z \in \mathbb{D}$. Therefore,

$$ (3.86) \qquad \mu^{(2)} \ge \frac{1}{2} \mu_{\text{pr}} = \frac{\text{cap}_{\mathbb{D}}(K_{\max})}{\text{cap}_{\mathbb{D}}(K_f)} \mu_{\mathbb{D}, K_{\max}} $$

which, upon recalling once again (2.5) and the fact that $\mu_{\mathbb{D}, K_{\max}}$ is a probability measure, gives us

$$ (3.87) \qquad \text{cap}_{\mathbb{D}}(K_{\max}) = \text{cap}_{\mathbb{D}}(K_f) \quad \text{and} \quad \mu^{(2)} = \mu_{\mathbb{D}, K_{\max}}. $$

In particular $\tilde{N}_2 = \emptyset$ and the first equality in (3.87), combined with the minimality and uniqueness of $K_f$, yields that $K_{\max} = K_f$. In addition, as $\|\mu^{(2)}\| = 1$ by (3.87), we get from (3.74) that $\text{supp}(\mu^{(1)}) \cap D_{\max} = \emptyset$ and therefore $\mu^{(1)} = \mu^{(2)}$, by (A.22), (3.71), and the fact that $\mathbb{D} \setminus D_{\max}$ is its own base (because so is $\mathcal{R} \setminus G_{\max}$ and we can use Lemma A.2). Moreover, we also get from (3.74) that
$\text{supp}_f(\mu^{(1)}) \cap D_{\text{max}} = \emptyset$ (note that $\text{supp}_f(\mu^{(1)})$ exists because $\mu^{(1)}$ is admissible, see Section A.9). Remembering that the fine open set $D'$ is regular by (3.66), we get from (A.24) and Lemma A.6 that $\text{supp}_f(\mu^{(1)}) \supset \cup_t \hat{\partial}_t D'_t$ where $D'_t$ are the finely connected components of $D'$ such that $\mu(D_t) \neq 0$. However, since $K_{\text{max}}(= K_f)$ has no fine interior by inspection of (2.17), each $\hat{\partial}_t D'_t$ must intersect $D_{\text{max}}$ whenever $D'_t$ is nonempty. Hence, $\mu$ cannot charge $D'$ as otherwise it would contradict that $\text{supp}_f(\mu^{(1)}) \cap D_{\text{max}} = \emptyset$. Consequently, since $D\setminus D'$ is its own base, we conclude in view of (A.22) and (3.67) that $\mu^{(1)} = \mu$. Thus, we obtain altogether that

$$ (3.88) \qquad \mu_{D, K_f} = \mu^{(2)} = \mu^{(1)} = \mu = \mu', $$

where the last equality comes from the inequalities $\mu \le \mu'$ and $\|\mu'\| \le 1$, see (3.27). In addition, since $\mu_{D, K_f}$ has finite potential everywhere, we get that $\tilde{N}_0 = N_0 = \emptyset$ whence also $\tilde{N}_1 = N_1 = \emptyset$.

From (3.88) one sees that $\mu$ does not charge $D_{\text{max}}$, implying in view of (3.44) that $\text{ler}$ is subharmonic on $V_{\text{max}} = p^{-1}(D_{\text{max}})$. In particular, since $g(\mu_{D, K_{\text{max}}}, D; \cdot)$ extends continuously by zero on $T$ and $h'$ by $2/\text{cap}_D(K_f)$ on $T$, see Lemma 3.2), while $\text{fine}\lim_{z \to \zeta} \text{ler}(z) = \text{ler}(\zeta) < 0$ when $\zeta \in J$ by the fine continuity of $\text{ler}$ and the fact that $J \subset G_-$, it follows from the relative fine boundary maximum principle [20, Theorem 10.8] that $\text{ler} < 0$ in the annular region $\mathcal{A}(T, J)$ bounded by $T$ and $J$, since it is bounded above by the fine potential $g(\hat{\mu}_{D, K_{\text{max}}}, \mathcal{R}; \cdot)$ there, see (A.34). Let now $G'$ be a finely connected component of $G_+$. Since $G^+ = p^{-1}(p(G^+))$ by Lemma 3.10 and $\mathcal{A}(T, J) \subset G_-$, $p(G')$ lies in the interior of $p(J)$. Hence, $D' = p(G') \in D_J$ and by what precedes $D' \cap K_f = \emptyset$ as otherwise $\mu = \mu_{D, K_f}$ would charge $D'$ because it is carried by the whole set $K_f$ according to (A.30), i.e., it cannot be carried by $K_f \setminus D'$ (which is finely closed). Consequently, $G' \cap p^{-1}(K_f) = \emptyset$ which implies that $\text{ler} \le 0$ on $p^{-1}(K_f)$, and the relative fine boundary maximum principle in turn implies that $\text{ler} \le 0$ on $V_{\text{max}}$. Moreover, $\text{ler} < 0$ on $G_{\text{max}}$ as it is strictly negative on $\mathcal{A}(T, J)$. Immediately we deduce that $G_+ = \emptyset$ and $G_{\text{max}} \subseteq G_-$. Furthermore, since $G_-$ lies schlicht over $D$ by Lemma 3.11 while $K_f$ has no fine interior, maximality of $G_{\text{max}}$ implies that $G_{\text{max}} = G_-$. Altogether, we obtain that $D' = V' = \emptyset$ and $G_- = G_{\text{max}}$. In particular, the step of Lemma 3.13 is vacuous and $v^{(1)} = v$ as well as $\text{ler} = \text{ler}^{(1)}$. Moreover, $\text{ler}(z) = 0$ holds for $z \in \mathcal{R}\setminus G_{\text{max}}$.

Next, by (3.86), (3.88) and the construction of the measures $\mu_{\text{pr}}$, $\mu_R$ in Lemma 3.17, along with the discussion after (3.81), one has

$$ 2\mu_{D, K_f} = 2\mu^{(2)} = \mu_{\text{pr}} = p_*(\mu_R) \quad \text{and} \quad \mu_R \le (\hat{\mu}_{D, K_f})_{\lfloor \partial G_{\text{max}}} $$

Lemma 3.16 now yields that this last inequality is in fact an equality. As $\mu_R - \nu_R$ is the Riesz charge of $\text{ler}^{(2)}$ that vanishes on $\mathcal{R}\setminus G_{\text{max}}$ by (3.73), the discussion after (3.81) implies $\nu_R = v^{(2)}_{\lfloor \partial G_{\text{max}}}$, and since $v_{\text{pr}} = p_*(v_R)$ we get from (3.84) and (3.71) that

$$ (3.89) \qquad 0 = v^{(2)}(\bar{G}_{\text{max}}) = v^{(1)}(\bar{G}_{\text{max}}) = v(\bar{G}_{\text{max}}), $$

where the middle equality holds by (A.24) and Lemma A.6 (because $\delta_z^{\mathcal{R}\setminus G_{\text{max}}}$ is a strictly positive function of $z$ in the regular open set $G_{\text{max}}$), while the last equality comes from $\text{ler} = \text{ler}^{(1)}$. Recall that $v = v'' + v' = v^* + \tilde{v} + v'$, see (3.46) and (3.38), where $v^*$ the vague limit point of $\{v_n\}$ in $\mathcal{R}$, see Lemma 3.5. As $\text{ler}(z) = 0$ for $z \in \mathcal{R}\setminus G_{\text{max}}$ and $\text{ler}$ is a $\delta$-subharmonic function, we get from [13, Theorem 2] that its Riesz charge is supported on $\hat{\partial}G_{\text{max}}$. In view of (3.89), it entails that

$$ (3.90) \qquad (\hat{\mu}_{D, K_f})_{\lfloor p^{-1}(K_f) \setminus \hat{\partial} G_{\text{max}}} = v = v'' = v^*, $$

where $\tilde{v} + v' = 0$ since it is a measure supported on $E_f$, see Lemma 3.1 and (3.36), while $v$ does not charge polar sets by the first equality above. Since $v = v^{(1)}$ and $v^{(1)}$ has no mass in $V_{\text{max}}$ by (3.90), the step of Lemma 3.15 was also vacuous. We thus get that $\text{ler} = \text{ler}^{(2)}$, and it follows from (3.84) together with the construction of $h_{\text{pr}}$ in Lemma 3.17 that $h_R = h'$. Therefore, the inequality in (3.73)
$\text{supp}_f(\mu^{(1)}) \cap D_{\max} = \emptyset$ (note that $\text{supp}_f(\mu^{(1)})$ exists because $\mu^{(1)}$ is admissible, see Section A.9). Remembering that the fine open set $D'$ is regular by (3.66), we get from (A.24) and Lemma A.6 that $\text{supp}_f(\mu^{(1)}) \supset \cup_t \hat{\partial}_t D'_t$ where $D'_t$ are the finely connected components of $D'$ such that $\mu(D_t) \neq 0$. However, since $K_{\max}(= K_f)$ has no fine interior by inspection of (2.17), each $\hat{\partial}_t D'_t$ must intersect $D_{\max}$ whenever $D'_t$ is nonempty. Hence, $\mu$ cannot charge $D'$ as otherwise it would contradict that $\text{supp}_f(\mu^{(1)}) \cap D_{\max} = \emptyset$. Consequently, since $D\setminus D'$ is its own base, we conclude in view of (A.22) and (3.67) that $\mu^{(1)} = \mu$. Thus, we obtain altogether that

$$ (3.88) \qquad \mu_{\mathbb{D},K_f} = \mu^{(2)} = \mu^{(1)} = \mu = \mu', $$

where the last equality comes from the inequalities $\mu \le \mu'$ and $\|\mu'\| \le 1$, see (3.27). In addition, since $\mu_{\mathbb{D},K_f}$ has finite potential everywhere, we get that $\tilde{N}_0 = N_0 = \emptyset$ whence also $\tilde{N}_1 = N_1 = \emptyset$.

From (3.88) one sees that $\mu$ does not charge $D_{\max}$, implying in view of (3.44) that $\text{ler}$ is subharmonic on $V_{\max} = p^{-1}(D_{\max})$. In particular, since $g(\mu_{\mathbb{D},K_{\max}}, \mathbb{D}; \cdot)$ extends continuously by zero on $\mathbb{T}$ and $h'$ by $2/\text{cap}_{\mathbb{D}}(K_f)$ on $\mathcal{T}$, see Lemma 3.2), while $\text{fine}\lim_{z \to \zeta} \text{ler}(z) = \text{ler}(\zeta) < 0$ when $\zeta \in J$ by the fine continuity of $\text{ler}$ and the fact that $J \subset G_-$, it follows from the relative fine boundary maximum principle [20, Theorem 10.8] that $\text{ler} < 0$ in the annular region $\mathcal{A}(\mathcal{T}, J)$ bounded by $\mathcal{T}$ and $J$, since it is bounded above by the fine potential $g(\hat{\mu}_{\mathbb{D},K_{\max}}, \mathcal{R}; \cdot)$ there, see (A.34). Let now $G'$ be a finely connected component of $G_+$. Since $G^+ = p^{-1}(p(G^+))$ by Lemma 3.10 and $\mathcal{A}(\mathcal{T}, J) \subset G_-$, $p(G')$ lies in the interior of $p(J)$. Hence, $D' = p(G') \in D_J$ and by what precedes $D' \cap K_f = \emptyset$ as otherwise $\mu = \mu_{\mathbb{D},K_f}$ would charge $D'$ because it is carried by the whole set $K_f$ according to (A.30), i.e., it cannot be carried by $K_f \setminus D'$ (which is finely closed). Consequently, $G' \cap p^{-1}(K_f) = \emptyset$ which implies that $\text{ler} \le 0$ on $p^{-1}(K_f)$, and the relative fine boundary maximum principle in turn implies that $\text{ler} \le 0$ on $V_{\max}$. Moreover, $\text{ler} < 0$ on $G_{\max}$ as it is strictly negative on $\mathcal{A}(\mathcal{T}, J)$. Immediately we deduce that $G_+ = \emptyset$ and $G_{\max} \subseteq G_-$. Furthermore, since $G_-$ lies schlicht over $\mathbb{D}$ by Lemma 3.11 while $K_f$ has no fine interior, maximality of $G_{\max}$ implies that $G_{\max} = G_-$. Altogether, we obtain that $D' = V' = \emptyset$ and $G_- = G_{\max}$. In particular, the step of Lemma 3.13 is vacuous and $v^{(1)} = v$ as well as $\text{ler} = \text{ler}^{(1)}$. Moreover, $\text{ler}(z) = 0$ holds for $z \in \mathbb{R}\setminus G_{\max}$.

Next, by (3.86), (3.88) and the construction of the measures $\mu_{\text{pr}}$, $\mu_{\mathbb{R}}$ in Lemma 3.17, along with the discussion after (3.81), one has

$$ 2\mu_{\mathbb{D},K_f} = 2\mu^{(2)} = \mu_{\text{pr}} = p_*(\mu_{\mathbb{R}}) \quad \text{and} \quad \mu_{\mathbb{R}} \le (\hat{\mu}_{\mathbb{D},K_f})_{\lfloor \partial G_{\max}}. $$

Lemma 3.16 now yields that this last inequality is in fact an equality. As $\mu_{\mathbb{R}} - \nu_{\mathbb{R}}$ is the Riesz charge of $\text{ler}^{(2)}$ that vanishes on $\mathbb{R}\setminus G_{\max}$ by (3.73), the discussion after (3.81) implies $\nu_{\mathbb{R}} = v^{(2)}_{\lfloor \partial G_{\max}}$, and since $v_{\text{pr}} = p_*(v_{\mathbb{R}})$ we get from (3.84) and (3.71) that

$$ (3.89) \qquad 0 = v^{(2)}(\bar{G}_{\max}) = v^{(1)}(\bar{G}_{\max}) = v(\bar{G}_{\max}), $$

where the middle equality holds by (A.24) and Lemma A.6 (because $\delta_z^{\mathbb{R}\setminus G_{\max}}$ is a strictly positive function of $z$ in the regular open set $G_{\max}$), while the last equality comes from $\text{ler} = \text{ler}^{(1)}$. Recall that $v = v'' + v' = v^* + \tilde{v} + v'$, see (3.46) and (3.38), where $v^*$ the vague limit point of $\{v_n\}$ in $\mathbb{R}$, see Lemma 3.5. As $\text{ler}(z) = 0$ for $z \in \mathbb{R}\setminus G_{\max}$ and $\text{ler}$ is a $\delta$-subharmonic function, we get from [13, Theorem 2] that its Riesz charge is supported on $\hat{\partial}G_{\max}$. In view of (3.89), it entails that

$$ (3.90) \qquad (\hat{\mu}_{\mathbb{D},K_f})_{\lfloor p^{-1}(K_f) \setminus \hat{\partial} G_{\max}} = v = v'' = v^*, $$

where $\tilde{v} + v' = 0$ since it is a measure supported on $E_f$, see Lemma 3.1 and (3.36), while $v$ does not charge polar sets by the first equality above. Since $v = v^{(1)}$ and $v^{(1)}$ has no mass in $V_{\max}$ by (3.90), the step of Lemma 3.15 was also vacuous. We thus get that $\text{ler} = \text{ler}^{(2)}$, and it follows from (3.84) together with the construction of $h_{\text{pr}}$ in Lemma 3.17 that $h_{\mathbb{R}} = h'$. Therefore, the inequality in (3.73)
is in fact an equality by Lemma 3.2, and consequently

$$ (3.91) \qquad \operatorname{ler}(z) = \begin{cases} 2g(\mu_{\mathbb{D},K_f}, \mathbb{D}; p(z)) - 2/\operatorname{cap}_{\mathbb{D}}(K_f), & z \in G_{\max}, \\ 0, & z \in \mathcal{R} \setminus G_{\max}, \end{cases} $$

as both sides of (3.91) are continuous on $\bar{\mathcal{R}}$, equal to zero on $\mathcal{R}\setminus G_{\max}$, equal to $-2/\operatorname{cap}_{\mathbb{D}}(K_f)$ on $\mathcal{T}$, and harmonic in $G_{\max}$ so that the equality in $G_{\max}$ is consequence of the maximum principle for harmonic functions.

Notice that we started our proof with the limit (3.7) taking place along the full sequence $\mathbb{N}$ of integers, that was later refined into a subsequence in Section 3.5, and refined still further in Section 3.7 to a subsequence $\mathcal{N}$ along which all the above results hold. However, we could have initiated our argument using any subsequence $\mathbb{N}_0 \subset \mathbb{N}$ in (3.7) with the same conclusions holding along some $\mathcal{N} \subseteq \mathbb{N}_0$. Hence, if there existed a subsequence along which either $\mu' \neq \mu_{\mathbb{D},K_f}$, $v' \neq 0$, $\tilde{v} \neq 0$, or $v^*$ are not as in the left-hand side of (3.90), or if $h''$ were not equal to zero, then we could use it as $\mathbb{N}_0$ in (3.7) to arrive at a contradiction. Hence, all lemmas of this section hold along the full sequence $\mathbb{N}$.

This proves (2.15) and the analogous assertion in Theorem 2.5.

### 3.11. Convergence in Capacity.
In view of Sections 3.2-3.3, to complete the proof of Theorems 2.1 and 2.5, we only need to establish that

$$ (3.92) \qquad \frac{1}{2n} \log |(f - N(M_n))(z)| \stackrel{\text{cap}}{\longrightarrow} g(\mu_{\mathbb{D},K_f}, \mathbb{D}; z) - \frac{1}{\text{cap}_{\mathbb{D}}(K_f)} \quad \text{as } n \to \infty, \quad z \in \mathbb{D} \setminus K_f. $$

Let $K \subset \mathbb{D} \setminus K_f$ be compact. Further, let $V$ be an open neighborhood of $p^{-1}(K_f)$ on $\mathcal{R}$ whose closure is disjoint from $K$, and $U$ an open subset of $\mathcal{R}$ containing $K$ whose closure is disjoint from $\bar{V} \cup \mathcal{T}$. For convenience, we also assume that $V = p^{-1}(p(V))$. Recall from (3.9)-(3.11) the relation

$$ \frac{1}{2n} \log |(f - N(M_n) \circ p)(z)| = \frac{1}{2} (h_{n,n}(z) + g(\mu_n, \mathbb{D}; p(z)) - g(v_n, \Omega_n; z)), \quad z \in \Omega_n. $$

We have established in the previous section that the functions $h_{n,n}$ converge to $-h_{\mathcal{R}} = -h'$ locally uniformly in $\mathcal{R}\setminus E_f$ and therefore on $K$, see Lemma 3.1 (note that $\mathcal{N}'$ may now be replaced by $\mathbb{N}$ by the last remark in Section 3.10). Since

$$ \frac{1}{2}(-h_{\mathcal{R}}(z) + g(\mu_{\mathbb{D},K_f}, \mathbb{D}; p(z)) - g(v, \mathcal{R}; z)) = \frac{1}{2}\operatorname{ler}(z) = g(\mu_{\mathbb{D},K_f}, \mathbb{D}; p(z)) - \frac{1}{\operatorname{cap}_{\mathbb{D}}(K_f)} $$

for $z \in G_{\max}$ by (3.46) and (3.91), to obtain (3.92) we only need to establish that for any $a > 0$

$$ (3.93) \qquad \left\{ \begin{array}{l} \lim_{n \to \infty} \operatorname{cap}_{\mathbb{D}} (\{z \in p(K) : |g(\mu_n, \mathbb{D}; z) - g(\mu_{\mathbb{D},K_f}, \mathbb{D}; z)| > a\}) = 0, \\ \\ \lim_{n \to \infty} \operatorname{cap}_{\mathbb{D}} (p\{z \in K : |g(v_n, \Omega_n; z) - g(v, \mathcal{R}; z)| > a\}) = 0. \end{array} \right. $$

Write $\mu_n = \mu_{n,1} + \mu_{n,2}$, where $\mu_{n,1} := \mu_{n|p(V)}$. Notice that $\mu_{n,1} \stackrel{*}{\longrightarrow} \mu_{\mathbb{D},K_f}$ and therefore the differences $g(\mu_{n,1}, \mathbb{D}; \cdot) - g(\mu_{\mathbb{D},K_f}, \mathbb{D}; \cdot)$ converge uniformly to zero on $K$. Recall that the Green potential of a measure supported in $\mathbb{D}$ can be written as a difference of the logarithmic potentials of that measure and of its balayage onto $\mathcal{T}$, see (A.29). In [6, Lemma 21], it was shown that when compactly supported measures converge weak* to the zero measure then their logarithmic potentials converge to zero in logarithmic capacity on $\mathbb{C}$, see (A.6) for the definition of logarithmic capacity. Thus, the first relation in (3.93) holds but in logarithmic capacity. However, since $t \mapsto g_D(t, z) + \log|z - t|$ is uniformly bounded for $z$ on compact subsets of $\mathbb{D}$, the convergence in logarithmic and Greenian capacities in $\mathbb{D}$ are equivalent. This finishes the proof of the first limit in (3.93).

The proof of the second limit in (3.93) proceeds similarly, but requires a more detailed analysis. As we have shown in the previous subsection, the measures $v_{n|\Omega_n}$ converge vaguely to $v^*$ on $\mathcal{R}$,
is in fact an equality by Lemma 3.2, and consequently

$$ (3.91) \qquad \operatorname{ler}(z) = \begin{cases} 2g(\mu_{\mathbb{D},K_f}, \mathbb{D}; p(z)) - 2/\operatorname{cap}_{\mathbb{D}}(K_f), & z \in G_{\max}, \\ 0, & z \in \mathcal{R} \setminus G_{\max}, \end{cases} $$

as both sides of (3.91) are continuous on $\bar{\mathcal{R}}$, equal to zero on $\mathcal{R}\setminus G_{\max}$, equal to $-2/\operatorname{cap}_{\mathbb{D}}(K_f)$ on $\mathcal{T}$, and harmonic in $G_{\max}$ so that the equality in $G_{\max}$ is consequence of the maximum principle for harmonic functions.

Notice that we started our proof with the limit (3.7) taking place along the full sequence $\mathbb{N}$ of integers, that was later refined into a subsequence in Section 3.5, and refined still further in Section 3.7 to a subsequence $\mathcal{N}$ along which all the above results hold. However, we could have initiated our argument using any subsequence $\mathbb{N}_0 \subset \mathbb{N}$ in (3.7) with the same conclusions holding along some $\mathcal{N} \subseteq \mathbb{N}_0$. Hence, if there existed a subsequence along which either $\mu' \neq \mu_{\mathbb{D},K_f}$, $v' \neq 0$, $\tilde{v} \neq 0$, or $v^*$ are not as in the left-hand side of (3.90), or if $h''$ were not equal to zero, then we could use it as $\mathbb{N}_0$ in (3.7) to arrive at a contradiction. Hence, all lemmas of this section hold along the full sequence $\mathbb{N}$.
This proves (2.15) and the analogous assertion in Theorem 2.5.

### 3.11. Convergence in Capacity.
In view of Sections 3.2-3.3, to complete the proof of Theorems 2.1 and 2.5, we only need to establish that

$$ (3.92) \qquad \frac{1}{2n} \log |(f - N(M_n))(z)| \xrightarrow{\text{cap}} g(\mu_{\mathbb{D},K_f}, \mathbb{D}; z) - \frac{1}{\operatorname{cap}_{\mathbb{D}}(K_f)} \quad \text{as } n \to \infty, \quad z \in \mathbb{D} \setminus K_f. $$

Let $K \subset \mathbb{D} \setminus K_f$ be compact. Further, let $V$ be an open neighborhood of $p^{-1}(K_f)$ on $\mathcal{R}$ whose closure is disjoint from $K$, and $U$ an open subset of $\mathcal{R}$ containing $K$ whose closure is disjoint from $\bar{V} \cup \mathcal{T}$. For convenience, we also assume that $V = p^{-1}(p(V))$. Recall from (3.9)-(3.11) the relation

$$ \frac{1}{2n} \log |(f - N(M_n) \circ p)(z)| = \frac{1}{2} (h_{n,n}(z) + g(\mu_n, \mathbb{D}; p(z)) - g(v_n, \Omega_n; z)), \quad z \in \Omega_n. $$

We have established in the previous section that the functions $h_{n,n}$ converge to $-h_{\mathcal{R}} = -h'$ locally uniformly in $\mathcal{R}\setminus E_f$ and therefore on $K$, see Lemma 3.1 (note that $\mathcal{N}'$ may now be replaced by $\mathbb{N}$ by the last remark in Section 3.10). Since

$$ \frac{1}{2}(-h_{\mathcal{R}}(z) + g(\mu_{\mathbb{D},K_f}, \mathbb{D}; p(z)) - g(v, \mathcal{R}; z)) = \frac{1}{2}\operatorname{ler}(z) = g(\mu_{\mathbb{D},K_f}, \mathbb{D}; p(z)) - \frac{1}{\operatorname{cap}_{\mathbb{D}}(K_f)} $$

for $z \in G_{\max}$ by (3.46) and (3.91), to obtain (3.92) we only need to establish that for any $a > 0$

$$ (3.93) \qquad \left\{ \begin{array}{l} \lim_{n \to \infty} \operatorname{cap}_{\mathbb{D}} (\{z \in p(K) : |g(\mu_n, \mathbb{D}; z) - g(\mu_{\mathbb{D},K_f}, \mathbb{D}; z)| > a\}) = 0, \\ \\ \lim_{n \to \infty} \operatorname{cap}_{\mathbb{D}} (p\{z \in K : |g(v_n, \Omega_n; z) - g(v, \mathcal{R}; z)| > a\}) = 0. \end{array} \right. $$

Write $\mu_n = \mu_{n,1} + \mu_{n,2}$, where $\mu_{n,1} := \mu_{n|p(V)}$. Notice that $\mu_{n,1} \xrightarrow{*} \mu_{\mathbb{D},K_f}$ and therefore the differences $g(\mu_{n,1}, \mathbb{D}; \cdot) - g(\mu_{\mathbb{D},K_f}, \mathbb{D}; \cdot)$ converge uniformly to zero on $K$. Recall that the Green potential of a measure supported in $\mathbb{D}$ can be written as a difference of the logarithmic potentials of that measure and of its balayage onto $\mathcal{T}$, see (A.29). In [6, Lemma 21], it was shown that when compactly supported measures converge weak* to the zero measure then their logarithmic potentials converge to zero in logarithmic capacity on $\mathbb{C}$, see (A.6) for the definition of logarithmic capacity. Thus, the first relation in (3.93) holds but in logarithmic capacity. However, since $t \mapsto g_D(t, z) + \log|z - t|$ is uniformly bounded for $z$ on compact subsets of $\mathbb{D}$, the convergence in logarithmic and Greenian capacities in $\mathbb{D}$ are equivalent. This finishes the proof of the first limit in (3.93).

The proof of the second limit in (3.93) proceeds similarly, but requires a more detailed analysis. As we have shown in the previous subsection, the measures $v_{n|\Omega_n}$ converge vaguely to $v^*$ on $\mathcal{R}$,
where $v^*$ stands for the left-hand side of (3.90). Hence, one has the limiting relation

$$ (3.94) \qquad v_{n,1} := v_{n|\overline{V} \cap \Omega_n} \xrightarrow{w^*} v^* $$

on $\mathcal{R}$. The functions $g_{\mathcal{R}}(z, w) - g_{\Omega_n}(z, w)$ uniformly converge to zero for $w \in V$ and $z \in K$, see (3.56) (we extend $g_{\Omega_n}(z, w)$ by zero to $\mathcal{R}\backslash\Omega_n$). Thus, as the measures $v_{n,1}$ have uniformly bounded masses (for they converge weak* to a finite measure), we get that

$$ (3.95) \qquad \lim_{n \to \infty} (g(v_{n,1}, \mathcal{R}; z) - g(v_{n,1}, \Omega_n; z)) = 0 $$

uniformly on $K$. Moreover, since the $g(v_{n,1}, \mathcal{R}; \cdot)$ are positive harmonic functions on $U$ for all $n$ large, they converge uniformly to $g(v^*, \mathcal{R}; \cdot)$ on $K$ (they converge pointwise on $U$ by (3.94) and hence uniformly on $K$ by Harnack's theorem). That is, we get from (3.95) that

$$ (3.96) \qquad \lim_{n \to \infty} g(v_{n,1}, \Omega_n; z) = g(v^*, \mathcal{R}, z) $$

uniformly on $K$. Next, let $v_{n,2} := v_{n|U}$. As the $v_n$ converge vaguely to $v^*$ and $\overline{U} \cap (\overline{V} \cup \mathcal{T}) = \emptyset$, the measures $v_{n,2}$ converge strongly to the zero measure. Subsequently, the measures $p_*(v_{n,2})$ strongly converge to zero. As before, it entails that the potentials $g(p_*(v_{n,2}), \mathcal{D}; p(\cdot))$ as well as $g(v_{n,2}, \mathcal{R}; \cdot)$ converge to zero in capacity on $K$, by (A.32). As the latter potentials majorize $g(v_{n,2}, \Omega_n; \cdot)$, these in turn converge to zero in capacity on $K$. Finally, define $v_{n,3} := v_n - v_{n,1} - v_{n,2}$. The potentials $g(v_{n,3}, \Omega_n; \cdot)$ form a sequence of positive harmonic functions in $U$. By Harnack's theorem there exists a subsequence of indices, say $\mathcal{N}_0$, along which these potentials converge locally uniformly on $U$ to some $h_0$ which is a non-negative harmonic function or $+\infty$. Now, we can initiate the proof of Theorem 2.5 in Section 3.5 with $\mathcal{N}_0$ instead of $\mathbb{N}$. Then, it would follow from (3.90) and (3.96) that the function $h''$ from Lemma 3.5 must coincide with $h_0$ in $U$. Proceeding with the remainder of the proof we again inevitably arrive at the conclusion that $h'' \equiv 0$. Hence, the potentials $g(v_{n,3}, \Omega_n; \cdot)$ converge to 0 locally uniformly on $U$ and thus uniformly on $K$. This finishes the proof of the second limit in (3.93) and, respectively, of (3.92); that is to say, of Theorem 2.5.

Finally, we need to account for Remark 2.2, in which we claimed that for any sequence $\{M_n\}$ of $n$-th root optimal meromorphic approximants to $f \in \mathcal{F}(\overline{\mathcal{C}}\backslash D)$ and any $\epsilon > 0$, it holds that

$$ (3.97) \qquad \lim_{n \to \infty} \operatorname{cap} \left\{ z \in K : \frac{1}{2n} \log |(f - M_n)(z)| - g(\mu_{D, K_f}, D; z) + \frac{1}{\operatorname{cap}_D(K_f)} > \epsilon \right\} = 0 $$

for every compact set $K \subset \overline{D}\backslash K_f$; moreover, if $|f - M_n|^{1/n}$ is asymptotically constant on $T$, i.e., if

$$ (3.98) \qquad \lim_{n \to \infty} \inf_{z \in \mathcal{T}} |(f - M_n)(z)|^{1/n} = \lim_{n \to \infty} \sup_{z \in \mathcal{T}} |(f - M_n)(z)|^{1/n} = \exp \left\{ - \frac{2}{\operatorname{cap}_D(K_f)} \right\}, $$

then the limit in (3.97) is supplemented by

$$ (3.99) \qquad \lim_{n \to \infty} \operatorname{cap} \left\{ z \in K : \frac{1}{2n} \log |(f - M_n)(z)| - g(\mu_{D, K_f}, D; z) + \frac{1}{\operatorname{cap}_D(K_f)} < -\epsilon \right\} = 0. $$

Together, (3.97) and (3.99) establish a full analog of (3.92) but in logarithmic rather than Greenian capacity, and not just in $D\backslash K_f$ but in $\overline{D}\backslash K_f$. Notice also that we work directly on the domain $D$ here, because logarithmic capacity is not preserved by conformal maps.

Since convergence in logarithmic and Greenian capacities are equivalent on compact subsets of $D\backslash K_f$, we only need to establish (3.97) and (3.99) for a compact set $K_\eta$ of the form

$$ K_\eta := \{z \in \overline{D} : 1 - \eta \le |\phi^{-1}(z)| \le 1\} $$
where $v^*$ stands for the left-hand side of (3.90). Hence, one has the limiting relation

$$ (3.94) \qquad v_{n,1} := v_{n|\overline{V} \cap \Omega_n} \xrightarrow{w^*} v^* $$

on $\mathcal{R}$. The functions $g_{\mathcal{R}}(z, w) - g_{\Omega_n}(z, w)$ uniformly converge to zero for $w \in V$ and $z \in K$, see (3.56) (we extend $g_{\Omega_n}(z, w)$ by zero to $\mathcal{R}\setminus\Omega_n$). Thus, as the measures $v_{n,1}$ have uniformly bounded masses (for they converge weak* to a finite measure), we get that

$$ (3.95) \qquad \lim_{n \to \infty} (g(v_{n,1}, \mathcal{R}; z) - g(v_{n,1}, \Omega_n; z)) = 0 $$

uniformly on $K$. Moreover, since the $g(v_{n,1}, \mathcal{R}; \cdot)$ are positive harmonic functions on $U$ for all $n$ large, they converge uniformly to $g(v^*, \mathcal{R}; \cdot)$ on $K$ (they converge pointwise on $U$ by (3.94) and hence uniformly on $K$ by Harnack's theorem). That is, we get from (3.95) that

$$ (3.96) \qquad \lim_{n \to \infty} g(v_{n,1}, \Omega_n; z) = g(v^*, \mathcal{R}, z) $$

uniformly on $K$. Next, let $v_{n,2} := v_{n|U}$. As the $v_n$ converge vaguely to $v^*$ and $\overline{U} \cap (\overline{V} \cup \mathcal{T}) = \emptyset$, the measures $v_{n,2}$ converge strongly to the zero measure. Subsequently, the measures $p_*(v_{n,2})$ strongly converge to zero. As before, it entails that the potentials $g(p_*(v_{n,2}), \mathcal{D}; p(\cdot))$ as well as $g(v_{n,2}, \mathcal{R}; \cdot)$ converge to zero in capacity on $K$, by (A.32). As the latter potentials majorize $g(v_{n,2}, \Omega_n; \cdot)$, these in turn converge to zero in capacity on $K$. Finally, define $v_{n,3} := v_n - v_{n,1} - v_{n,2}$. The potentials $g(v_{n,3}, \Omega_n; \cdot)$ form a sequence of positive harmonic functions in $U$. By Harnack's theorem there exists a subsequence of indices, say $\mathcal{N}_0$, along which these potentials converge locally uniformly on $U$ to some $h_0$ which is a non-negative harmonic function or $+\infty$. Now, we can initiate the proof of Theorem 2.5 in Section 3.5 with $\mathcal{N}_0$ instead of $\mathbb{N}$. Then, it would follow from (3.90) and (3.96) that the function $h''$ from Lemma 3.5 must coincide with $h_0$ in $U$. Proceeding with the remainder of the proof we again inevitably arrive at the conclusion that $h'' \equiv 0$. Hence, the potentials $g(v_{n,3}, \Omega_n; \cdot)$ converge to 0 locally uniformly on $U$ and thus uniformly on $K$. This finishes the proof of the second limit in (3.93) and, respectively, of (3.92); that is to say, of Theorem 2.5.

Finally, we need to account for Remark 2.2, in which we claimed that for any sequence $\{M_n\}$ of $n$-th root optimal meromorphic approximants to $f \in \mathcal{F}(\overline{\mathcal{C}}\setminus D)$ and any $\epsilon > 0$, it holds that

$$ (3.97) \qquad \lim_{n \to \infty} \operatorname{cap} \left\{ z \in K : \frac{1}{2n} \log |(f - M_n)(z)| - g(\mu_{D, K_f}, D; z) + \frac{1}{\operatorname{cap}_D(K_f)} > \epsilon \right\} = 0 $$

for every compact set $K \subset \overline{D}\setminus K_f$; moreover, if $|f - M_n|^{1/n}$ is asymptotically constant on $T$, i.e., if

$$ (3.98) \qquad \lim_{n \to \infty} \inf_{z \in \mathcal{T}} |(f - M_n)(z)|^{1/n} = \lim_{n \to \infty} \sup_{z \in \mathcal{T}} |(f - M_n)(z)|^{1/n} = \exp \left\{ -\frac{2}{\operatorname{cap}_D(K_f)} \right\}, $$

then the limit in (3.97) is supplemented by

$$ (3.99) \qquad \lim_{n \to \infty} \operatorname{cap} \left\{ z \in K : \frac{1}{2n} \log |(f - M_n)(z)| - g(\mu_{D, K_f}, D; z) + \frac{1}{\operatorname{cap}_D(K_f)} < -\epsilon \right\} = 0. $$

Together, (3.97) and (3.99) establish a full analog of (3.92) but in logarithmic rather than Greenian capacity, and not just in $D\setminus K_f$ but in $\overline{D}\setminus K_f$. Notice also that we work directly on the domain $D$ here, because logarithmic capacity is not preserved by conformal maps.

Since convergence in logarithmic and Greenian capacities are equivalent on compact subsets of $D\setminus K_f$, we only need to establish (3.97) and (3.99) for a compact set $K_\eta$ of the form

$$ K_\eta := \{z \in \overline{D} : 1 - \eta \le |\phi^{-1}(z)| \le 1\} $$
for some $\eta > 0$ sufficiently small, where $\phi: \mathbb{D} \to D$ is a conformal map. Let $K'_\eta = \overline{K_{2\eta} \setminus K_\eta}$. We
claim that for all $n$ large enough there exists $r_n \in (1-4\eta, 1-2\eta)$ such that

$$
(3.100) \quad \left| \frac{1}{2n} \log |(f - M_n)(z)| - g(\mu_{D,K_f}, D; z) + \frac{1}{\mathrm{cap}_D(K_f)} \right| \le \epsilon,
$$

for all $z \in T_{r_n}$, where $T_r := \phi(\mathbb{T}_r)$. Indeed, let $E_{n,2\eta}$ be the set of all $z \in K'_{2\eta}$ for which the above inequality is false. As we have already shown, (3.92) holds for the $n$-th root optimal approximants $M_n$, both in Greenian and logarithmic capacities on compact subsets of $D \setminus K_f$, in particular, on $K'_{2\eta}$. Hence, $\mathrm{cap}(E_{n,2\eta}) \to 0$ as $n \to \infty$. On the other hand, if our claim were false, then $E_{n,2\eta} \cap T_r \neq \emptyset$ for every $r \in [1-4\eta, 1-2\eta]$. In this case it must hold that

$$
\mathrm{cap}(E_{n,\eta}) \geq \frac{\mathrm{cap}([1-4\eta, 1-2\eta])}{\| (\phi^{-1})' \|_{K'_\eta}} = \frac{\eta}{2\| (\phi^{-1})' \|_{K'_\eta}}
$$

because contractive maps, in particular $\phi_{[K'_\eta}^{-1}/\|(\phi^{-1})'\|_{K'_\eta}$ followed by the circular projection, do
not increase logarithmic capacity [53, Theorem 5.3.1]. The above contradiction proves our claim.
Clearly, $T_{r_n}$ is a Jordan curve homotopic to $T$ in $D\setminus K_f$ for all $\eta$ small enough. By adding the same
constant to $f$ and $M_n$ we can assume that $f$ is not vanishing on $K'_{2\eta}$. Hence, we deduce from (3.100)
that $|f-M_n|<|f|$ on $T_{r_n}$ for all $n$ large enough. Besides, the same inequality certainly holds on $T$
and therefore, by Rouché's theorem, the number of zeros of $f-M_n$ in $K_{2\eta}$ is less that the number
of its poles plus the degree of $f$ on $T \cup T_{r_n}$ (which is bounded by a constant independently of $n$).
It now follows from Theorem 2.5 that $f-M_n$ has little $o(1)$ poles and zeros in $K_{2\eta}$ as $n \to \infty$.
Denote by $b_{n,\eta}^{\text{pole}}$ and $b_{n,\eta}^{\text{zero}}$ the Blaschke products in $D$ vanishing at the poles and zeros of $f-M_n$
in $K_{2\eta}$, respectively. As Green potentials of measures converging strongly to zero converge to zero
in capacity in $D$, the functions

$$
e_n(z) := \frac{1}{2n} \log |(f - M_n)(z) b_{n,\eta}^{\text{pole}}(z) / b_{n,\eta}^{\text{zero}}(z)| - g(\mu_{D,K_f}, D; z) + \frac{1}{\text{cap}_D(K_f)}
$$

still converge to zero in capacity in $D \setminus K_f$. Hence, similarly to (3.100), we can find $r_n^* \in (1-2\eta, 1-\eta)$ such that $|e_n(z)| < \epsilon$ for all $z \in T_{r_n^*}$. As $\lim \sup_n \sup_{z \in T} e_n(z) \le 0$ by the $n$-th root optimality of $e_n$ and $\lim_n \|e_n\|_T = 0$ if in addition (3.98) holds (asymptotic circularity of the error), the desired conclusion (3.97) and (3.99) come out by applying the maximum principle for the harmonic function to $e_n$ on $K_\eta$, along with the minimum principle if (3.98) holds.

4. PROOF OF THEOREM 2.6

Since the considerations of Sections 3.2 and 3.3 still apply, we can assume that $D = \mathbb{D}$, i.e., that $f$
is analytic in $\bar{C} \setminus E$, where $E \subset \mathbb{D}$ is closed polar, and we may replace $\{M_n\}$ by the sequence $\{N(M_n)\}$
of its Nehari modifications. Write $N(M_n) = h_n/b_n$, where $h_n \in \mathcal{A}(\mathbb{D})$ and $b_n$ is a Blaschke product
vanishing at the poles of $N(M_n)$ according to their multiplicities. That is, $b_n = q_n/\tilde{q}_n$, where
$q_n \in \mathcal{M}_n(\mathbb{D})$ is a polynomial such that $q_n M_n \in \mathcal{A}(\mathbb{D})$ and $\tilde{q}_n(z) := z^n q_n(1/\bar{z})$ is the reciprocal
polynomial. We have that

$$
|f(z) - (h_n/b_n)(z)| = |(b_n f)(z) - h_n(z)|, \quad z \in \mathbb{T},
$$

since Blaschke products are unimodular on the unit circle. Thus, $h_n$ is in fact the best Nehari
approximant of $b_n f$ in $\mathcal{A}(\mathbb{D})$. Recall that the above expressions converge to zero faster than
geometrically to zero by the very choice of $\{M_n\}$.

Our goal is to show that $h_n/b_n$ converge in logarithmic capacity to $f$ in $\bar{D}\setminus E$ at faster than
geometric rate. That is, we fix a compact set $K \subset \bar{D}$ disjoint from $E$ and we will prove that

$$
(4.1) \quad \lim_{n \to \infty} \mathrm{cap}(\{z \in K : |e_n(z)| > a^n\}) = 0 \quad \text{for any } a > 0,
$$
for some $\eta > 0$ sufficiently small, where $\phi: \mathbb{D} \to D$ is a conformal map. Let $K'_\eta = \overline{K_{2\eta} \setminus K_\eta}$. We
claim that for all $n$ large enough there exists $r_n \in (1 - 4\eta, 1 - 2\eta)$ such that

$$
(3.100) \quad \left| \frac{1}{2n} \log |(f - M_n)(z)| - g(\mu_{D,K_f}, D; z) + \frac{1}{\mathrm{cap}_D(K_f)} \right| \le \epsilon,
$$

for all $z \in T_{r_n}$, where $T_r := \phi(\mathbb{T}_r)$. Indeed, let $E_{n,2\eta}$ be the set of all $z \in K'_{2\eta}$ for which the above inequality is false. As we have already shown, (3.92) holds for the $n$-th root optimal approximants $M_n$, both in Greenian and logarithmic capacities on compact subsets of $D \setminus K_f$, in particular, on $K'_{2\eta}$. Hence, $\mathrm{cap}(E_{n,2\eta}) \to 0$ as $n \to \infty$. On the other hand, if our claim were false, then $E_{n,2\eta} \cap T_r \neq \emptyset$ for every $r \in [1-4\eta, 1-2\eta]$. In this case it must hold that

$$
\mathrm{cap}(E_{n,\eta}) \ge \frac{\mathrm{cap}([1-4\eta, 1-2\eta])}{\| (\phi^{-1})' \|_{K'_\eta}} = \frac{\eta}{2\| (\phi^{-1})' \|_{K'_\eta}}
$$

because contractive maps, in particular $\phi_{[K'_\eta}^{-1}/\|(\phi^{-1})'\|_{K'_\eta}$ followed by the circular projection, do
not increase logarithmic capacity [53, Theorem 5.3.1]. The above contradiction proves our claim.
Clearly, $T_{r_n}$ is a Jordan curve homotopic to $T$ in $D\setminus K_f$ for all $\eta$ small enough. By adding the same
constant to $f$ and $M_n$ we can assume that $f$ is not vanishing on $K'_{2\eta}$. Hence, we deduce from (3.100)
that $|f-M_n| < |f|$ on $T_{r_n}$ for all $n$ large enough. Besides, the same inequality certainly holds on $T$
and therefore, by Rouché's theorem, the number of zeros of $f-M_n$ in $K_{2\eta}$ is less that the number
of its poles plus the degree of $f$ on $T \cup T_{r_n}$ (which is bounded by a constant independently of $n$).
It now follows from Theorem 2.5 that $f-M_n$ has little $o(1)$ poles and zeros in $K_{2\eta}$ as $n \to \infty$.
Denote by $b_{n,\eta}^{\text{pole}}$ and $b_{n,\eta}^{\text{zero}}$ the Blaschke products in $D$ vanishing at the poles and zeros of $f-M_n$
in $K_{2\eta}$, respectively. As Green potentials of measures converging strongly to zero converge to zero
in capacity in $D$, the functions

$$
e_n(z) := \frac{1}{2n} \log |(f - M_n)(z) b_{n,\eta}^{\text{pole}}(z) / b_{n,\eta}^{\text{zero}}(z)| - g(\mu_{D,K_f}, D; z) + \frac{1}{\text{cap}_D(K_f)}
$$

still converge to zero in capacity in $D \setminus K_f$. Hence, similarly to (3.100), we can find $r_n^* \in (1-2\eta, 1-\eta)$ such that $|e_n(z)| < \epsilon$ for all $z \in T_{r_n^*}$. As $\limsup_n \sup_{z \in T} e_n(z) \le 0$ by the $n$-th root optimality of $e_n$ and $\lim_n \|e_n\|_T = 0$ if in addition (3.98) holds (asymptotic circularity of the error), the desired conclusion (3.97) and (3.99) come out by applying the maximum principle for the harmonic function to $e_n$ on $K_\eta$, along with the minimum principle if (3.98) holds.

4. PROOF OF THEOREM 2.6

Since the considerations of Sections 3.2 and 3.3 still apply, we can assume that $D = \mathbb{D}$, i.e., that $f$
is analytic in $\bar{\mathbb{C}} \setminus E$, where $E \subset \mathbb{D}$ is closed polar, and we may replace $\{M_n\}$ by the sequence $\{N(M_n)\}$
of its Nehari modifications. Write $N(M_n) = h_n/b_n$, where $h_n \in \mathcal{A}(\mathbb{D})$ and $b_n$ is a Blaschke product
vanishing at the poles of $N(M_n)$ according to their multiplicities. That is, $b_n = q_n/\tilde{q}_n$, where
$q_n \in \mathcal{M}_n(\mathbb{D})$ is a polynomial such that $q_n M_n \in \mathcal{A}(\mathbb{D})$ and $\tilde{q}_n(z) := z^n q_n(1/\bar{z})$ is the reciprocal
polynomial. We have that

$$
|f(z) - (h_n/b_n)(z)| = |(b_n f)(z) - h_n(z)|, \quad z \in \mathbb{T},
$$

since Blaschke products are unimodular on the unit circle. Thus, $h_n$ is in fact the best Nehari approximant of $b_n f$ in $\mathcal{A}(\mathbb{D})$. Recall that the above expressions converge to zero faster than geometrically to zero by the very choice of $\{M_n\}$.

Our goal is to show that $h_n/b_n$ converge in logarithmic capacity to $f$ in $\bar{\mathbb{D}}\setminus E$ at faster than
geometric rate. That is, we fix a compact set $K \subset \bar{\mathbb{D}}$ disjoint from $E$ and we will prove that

$$
(4.1) \quad \lim_{n \to \infty} \operatorname{cap}(\{z \in K : |e_n(z)| > a^n\}) = 0 \quad \text{for any } a > 0,
$$
where $e_n(z) := f(z) - (h_n/b_n)(z)$ is the approximation error. This will establish convergence in logarithmic capacity on compact subsets of $\mathbb{D}\setminus E$. Subsequently, as we pointed out in Section 3.11, (4.1) yields an analogous claim for the Greenian capacity on any compact subset of $\mathbb{D}\setminus E$.

Since cap($E$) = 0, it follows from [53, Theorems 5.5.2 & 5.5.4] that for each $\eta > 0$ there is $k \in \mathbb{N}$ and $p_k \in \mathcal{M}_k(E)$ (we can take $p_k$ to be the $k$-th Fekete polynomial for $E$) such that

$$ (4.2) \qquad E \subset L_{\eta} := \{\zeta \in \mathbb{C} : |p_k(\zeta)| < \eta^k\}. $$

Pick $\eta < \min\{\text{dist}(K, E), 1\}$ to be adjusted later. Of necessity $K \cap L_{\eta} = \emptyset$, because $|p_k(z)| \ge \text{dist}(K, E)^k$ for $z \in K$. Let $\gamma \subset L_{\eta}$ be a system of closed curves encompassing each point of $E$ exactly once, and such that $\text{dist}(\gamma, E) < \text{dist}(\mathbb{T}, E)/4$. Then, one has that

$$ (4.3) \qquad f(z) = \int_{\gamma} \frac{f(\xi)}{z - \xi} \frac{d\xi}{2\pi i}, \quad z \in \bar{\mathbb{C}}\setminus\overline{\text{int } \gamma}, $$

by the Cauchy formula, where $\text{int } \gamma$ is the union of the bounded components of the complement of $\gamma$. Since $\mathbb{P}_-$ evaluated at $|z| > 1$ coincides with the Cauchy projection having kernel $(2\pi i(z - \zeta))^{-1}d\zeta$ on $\mathbb{T}$, the Hankel operator $\Gamma_{b_n f}$ acts on $v \in H^2$ by

$$ (4.4) \qquad \Gamma_{b_n f}(v)(z) = \int_{\mathbb{T}} \frac{b_n(\zeta)f(\zeta)v(\zeta)}{z - \zeta} \frac{d\zeta}{2\pi i}, \quad |z| > 1. $$

Inserting (4.3) into (4.4) yields by Fubini's theorem and the Cauchy formula for $H^2$-functions that

$$ (4.5) \qquad \Gamma_{b_n f}(v)(z) = \int_{\gamma} \frac{b_n(\xi)f(\xi)v(\xi)}{z - \xi} \frac{d\xi}{2\pi i}, \quad z \in \bar{\mathbb{C}}\setminus\overline{\text{int } \gamma}. $$

Observe that (4.5), initially proven for $|z| > 1$, actually defines an analytic extension to the exterior of $\gamma$ of $\Gamma_{b_n f}(v) = \mathbb{P}_-(b_n f v) \in H^2_-$ (and therefore an extension to $\bar{\mathbb{C}}\setminus E$ since $\gamma$ could be taken arbitrary close to $E$). Next, recall that a first singular vector of the Hankel operator $\Gamma_{b_n f}$ is an element $v_0 \in H^2$ of unit norm that maximizes $\|\Gamma_{b_n f}(v)\|$ over all $v \in H^2$ with $\|v\|_2 = 1$, and that it always can be chosen to be outer<fnref n="3" />, see for instance [5, p. 62]. Then, we get from (4.5) and (2.26) (applied with $n = 0$ and $f$ replaced by $fb_n$) that

$$ (4.6) \qquad (e_n b_n)(z) = (b_n f - h_n)(z) = \frac{1}{v_0(z)} \int_{\gamma} \frac{b_n(\xi)f(\xi)v_0(\xi)}{z - \xi} \frac{d\xi}{2\pi i}, \quad z \in \bar{\mathbb{D}}\setminus\overline{\text{int } \gamma}. $$

Note that the right-hand side of (4.6) is analytic in $\bar{\mathbb{D}}\setminus\overline{\text{int } \gamma}$, since $v_0$ is outer and thus has no zeros in $\mathbb{D}$. Let $B_k := p_k/\tilde{p}_k$. Similarly to (4.3)–(4.6), it holds that

$$ (4.7) \qquad \mathbb{P}_- (e_n b_n B_k^\ell)(z) = \frac{1}{2\pi i} \int_{\gamma} \frac{b_n(\xi)f(\xi)B_k^\ell(\xi)}{z - \xi} d\xi, \quad z \in \bar{\mathbb{D}}\setminus\overline{\text{int } \gamma}, $$

where $\ell \in \mathbb{N}$ is such that $\ell k \le n < (\ell + 1)k$ and the right-hand side again defines an analytic extension of the left-hand side into the exterior of $\gamma$. Indeed, we can express the left-hand side of (4.7) for $|z| > 1$ as the Cauchy integral of $e_n b_n B_k^\ell$ on $\mathbb{T}$ like we did in (4.4) for $\mathbb{P}_-(b_n f v)$. Since $e_n b_n$ is analytic across $\mathbb{T}$ and $B_k \in \mathcal{H}(\bar{\mathbb{D}})$, we then deform the contour of integration into a circle of radius slightly smaller than 1, which can be done without changing the value of the integral by Cauchy's theorem. Subsequently, we insert (4.6) in this integral and use Fubini's theorem and the residue formula as before to get (4.7).

Recall that $\text{dist}(\mathbb{T}, E) \ge \text{dist}(K, E) > \eta$ by construction. Observe also that $|p_k| \le \eta^k$ on $\gamma$ by (4.2) and that $|\tilde{p}_k| \ge \text{dist}(\mathbb{T}, E)^k$ in $\mathbb{D}$. Since $|b_n| \le 1$, we get from our choice of $\ell$ that

$$ (4.8) \qquad |\mathbb{P}_- (e_n b_n B_k^\ell)(z)| \le \left( \frac{\eta}{\text{dist}(\mathbb{T}, E)} \right)^{n-k} \frac{|\gamma| \|f\|_\gamma}{\text{dist}(z, \gamma)}, \quad z \in \bar{\mathbb{D}}\setminus\overline{\text{int } \gamma}, $$

<footnote n="3">³An outer function $w \in H^2$ is of the form $w(z) = \alpha \exp \left\{ \int_{\mathbb{T}} \frac{\xi+z}{\xi-z} \log |w(\xi)| \frac{|d\xi|}{2\pi} \right\}$, with $w_{|\mathbb{T}} \in L^2(\mathbb{T})$ and $|\alpha| = 1$.</footnote>
where $e_n(z) := f(z) - (h_n/b_n)(z)$ is the approximation error. This will establish convergence in logarithmic capacity on compact subsets of $\mathbb{D}\setminus E$. Subsequently, as we pointed out in Section 3.11, (4.1) yields an analogous claim for the Greenian capacity on any compact subset of $\mathbb{D}\setminus E$.

Since cap($E$) = 0, it follows from [53, Theorems 5.5.2 & 5.5.4] that for each $\eta > 0$ there is $k \in \mathbb{N}$ and $p_k \in \mathcal{M}_k(E)$ (we can take $p_k$ to be the $k$-th Fekete polynomial for $E$) such that

$$ (4.2) \qquad E \subset L_{\eta} := \{\zeta \in \mathbb{C} : |p_k(\zeta)| < \eta^k\}. $$

Pick $\eta < \min\{\text{dist}(K, E), 1\}$ to be adjusted later. Of necessity $K \cap L_{\eta} = \emptyset$, because $|p_k(z)| \ge \text{dist}(K, E)^k$ for $z \in K$. Let $\gamma \subset L_{\eta}$ be a system of closed curves encompassing each point of $E$ exactly once, and such that $\text{dist}(\gamma, E) < \text{dist}(\mathbb{T}, E)/4$. Then, one has that

$$ (4.3) \qquad f(z) = \int_{\gamma} \frac{f(\xi)}{z - \xi} \frac{d\xi}{2\pi i}, \quad z \in \bar{\mathbb{C}}\setminus\overline{\text{int } \gamma}, $$

by the Cauchy formula, where $\text{int } \gamma$ is the union of the bounded components of the complement of $\gamma$. Since $\mathbb{P}_-$ evaluated at $|z| > 1$ coincides with the Cauchy projection having kernel $(2\pi i(z - \zeta))^{-1}d\zeta$ on $\mathbb{T}$, the Hankel operator $\Gamma_{b_n f}$ acts on $v \in H^2$ by

$$ (4.4) \qquad \Gamma_{b_n f}(v)(z) = \int_{\mathbb{T}} \frac{b_n(\zeta)f(\zeta)v(\zeta)}{z - \zeta} \frac{d\zeta}{2\pi i}, \quad |z| > 1. $$

Inserting (4.3) into (4.4) yields by Fubini's theorem and the Cauchy formula for $H^2$-functions that

$$ (4.5) \qquad \Gamma_{b_n f}(v)(z) = \int_{\gamma} \frac{b_n(\xi)f(\xi)v(\xi)}{z - \xi} \frac{d\xi}{2\pi i}, \quad z \in \bar{\mathbb{C}}\setminus\overline{\text{int } \gamma}. $$

Observe that (4.5), initially proven for $|z| > 1$, actually defines an analytic extension to the exterior of $\gamma$ of $\Gamma_{b_n f}(v) = \mathbb{P}_-(b_n f v) \in H^2_-$ (and therefore an extension to $\bar{\mathbb{C}}\setminus E$ since $\gamma$ could be taken arbitrary close to $E$). Next, recall that a first singular vector of the Hankel operator $\Gamma_{b_n f}$ is an element $v_0 \in H^2$ of unit norm that maximizes $\|\Gamma_{b_n f}(v)\|$ over all $v \in H^2$ with $\|v\|_2 = 1$, and that it always can be chosen to be outer<fnref n="3" />, see for instance [5, p. 62]. Then, we get from (4.5) and (2.26) (applied with $n = 0$ and $f$ replaced by $fb_n$) that

$$ (4.6) \qquad (e_n b_n)(z) = (b_n f - h_n)(z) = \frac{1}{v_0(z)} \int_{\gamma} \frac{b_n(\xi)f(\xi)v_0(\xi)}{z - \xi} \frac{d\xi}{2\pi i}, \quad z \in \bar{\mathbb{D}}\setminus\overline{\text{int } \gamma}. $$

Note that the right-hand side of (4.6) is analytic in $\bar{\mathbb{D}}\setminus\overline{\text{int } \gamma}$, since $v_0$ is outer and thus has no zeros in $\mathbb{D}$. Let $B_k := p_k/\tilde{p}_k$. Similarly to (4.3)–(4.6), it holds that

$$ (4.7) \qquad \mathbb{P}_- (e_n b_n B_k^\ell)(z) = \frac{1}{2\pi i} \int_{\gamma} \frac{b_n(\xi) f(\xi) B_k^\ell(\xi)}{z - \xi} d\xi, \quad z \in \bar{\mathbb{D}}\setminus\overline{\text{int } \gamma}, $$

where $\ell \in \mathbb{N}$ is such that $\ell k \le n < (\ell + 1)k$ and the right-hand side again defines an analytic extension of the left-hand side into the exterior of $\gamma$. Indeed, we can express the left-hand side of (4.7) for $|z| > 1$ as the Cauchy integral of $e_n b_n B_k^\ell$ on $\mathbb{T}$ like we did in (4.4) for $\mathbb{P}_-(b_n f v)$. Since $e_n b_n$ is analytic across $\mathbb{T}$ and $B_k \in \mathcal{H}(\bar{\mathbb{D}})$, we then deform the contour of integration into a circle of radius slightly smaller than 1, which can be done without changing the value of the integral by Cauchy's theorem. Subsequently, we insert (4.6) in this integral and use Fubini's theorem and the residue formula as before to get (4.7).

Recall that $\text{dist}(\mathbb{T}, E) \ge \text{dist}(K, E) > \eta$ by construction. Observe also that $|p_k| \le \eta^k$ on $\gamma$ by (4.2) and that $|\tilde{p}_k| \ge \text{dist}(\mathbb{T}, E)^k$ in $\mathbb{D}$. Since $|b_n| \le 1$, we get from our choice of $\ell$ that

$$ (4.8) \qquad |\mathbb{P}_- (e_n b_n B_k^\ell)(z)| \le \left( \frac{\eta}{\text{dist}(\mathbb{T}, E)} \right)^{n-k} \frac{|\gamma| \|f\|_\gamma}{\text{dist}(z, \gamma)}, \quad z \in \bar{\mathbb{D}}\setminus\overline{\text{int } \gamma}, $$

<footnote n="3">³An outer function $w \in H^2$ is of the form $w(z) = \alpha \exp\left\{\int_{\mathbb{T}} \frac{\xi+z}{\xi-z} \log|w(\xi)| \frac{|d\xi|}{2\pi}\right\}$, with $w|_{\mathbb{T}} \in L^2(\mathbb{T})$ and $|\alpha| = 1$.</footnote>
where $|\gamma|$ stands for the arclength of $\gamma$. In another connection, it follows from [49, Lemma] that for any $\varepsilon \in (0, 1/3)$ there exists $W_n \subset \mathbb{D}$ such that cap$(W_n) \le 3\varepsilon$ and

$$|p_k^\ell(\zeta)q_n(\zeta)| > \varepsilon^{n+\ell k} \|p_k^\ell q_n\|_{\mathbb{T}}, \quad \zeta \in \mathbb{D}\setminus W_n.$$

As $|\tilde{p}_k^\ell \tilde{q}_n|(z) \le \|\tilde{p}_k^\ell \tilde{q}_n\|_{\mathbb{T}} = \|p_k^\ell q_n\|_{\mathbb{T}}$ for $z \in \mathbb{D}$ by the maximum principle and the definition of the reciprocal polynomial, we get that $|(b_n B_k^\ell)(z)| \ge \varepsilon^{n+\ell k}$ for $z \in \mathbb{D}\setminus W_n$. Since $P_+ + P_-$ is the identity operator, we can use the analytic continuation provided by (4.7) to write

$$(4.9) \qquad e_n(z) = \frac{P_+(e_n b_n B_k^\ell)(z) + P_-(e_n b_n B_k^\ell)(z)}{b_n(z) B_k^\ell(z)}, \quad z \in \bar{\mathbb{D}}\setminus \overline{\text{int } \gamma}.$$

The estimate $|(b_n B_k^\ell)(z)| \ge \varepsilon^{n+\ell k}$ for $z \in \mathbb{D}\setminus W_n$, (4.8), and the definition of $\ell$ give us that

$$(4.10) \qquad \left| \frac{P_-(e_n b_n B_k^\ell)(z)}{b_n(z) B_k^\ell(z)} \right| \le \frac{1}{\varepsilon^{2n}} \left( \frac{\eta}{\text{dist}(\mathbb{T}, E)} \right)^{n-k} \frac{|\gamma| \|f\|_{\gamma}}{\text{dist}(z, \gamma)}, \quad z \in \bar{\mathbb{D}}\setminus(\overline{\text{int } \gamma} \cup W_n).$$

Now, given $0 < \varepsilon < 1$ and $0 < a < 1$, choose $\eta$ in (4.2) so that $0 < \eta < a\varepsilon^2\text{dist}(\mathbb{T}, E)$. Choice of $\eta$ of course fixes $k$ in (4.2). Then, since $K$ lies exterior to $\gamma$ because $K \cap L_\eta = \emptyset$, we get from (4.10) that there exists a natural number $n_0 = n_0(f, K, \varepsilon, \eta, \gamma)$ for which

$$(4.11) \qquad \left| \frac{P_-(e_n b_n B_k^\ell)(z)}{b_n(z) B_k^\ell(z)} \right| < a^n, \quad n \ge n_0, \quad z \in K \setminus W_n.$$

Next, as $\|e_n b_n B_k^\ell\|_{\mathbb{T}} = \|e_n\|_{\mathbb{T}} \to 0$ faster than geometrically with $n$ by hypothesis, (4.8) and the triangle inequality yield that

$$(4.12) \qquad \|P_+(e_n b_n B_k^\ell)\|_{\mathbb{T}} = \|e_n b_n B_k^\ell - P_-(e_n b_n B_k^\ell)\|_{\mathbb{T}} \le C \left( \frac{\eta}{\text{dist}(\mathbb{T}, E)} \right)^n, \quad n \ge n_0,$$

for some constant $C = C(f, E, \gamma)$ and some $n_0'$ depending on $C$ and the speed of approximation of $f$ by $h_n/b_n$. Subsequently, as in (4.11), we get from the estimate $|(b_n B_k^\ell)(z)| \ge \varepsilon^{n+\ell k}$ for $z \in \mathbb{D}\setminus W_n$, (4.12), and the maximum modulus principle that

$$(4.13) \qquad \left| \frac{P_+(e_n b_n B_k^\ell)(z)}{b_n(z) B_k^\ell(z)} \right| < a^n, \quad n \ge n_0'', \quad z \in K \setminus W_n,$$

for some natural number $n_0'' = n_0''(f, K, \varepsilon, \eta, \gamma)$. Because $\varepsilon$ and $a$ can be arbitrarily small and cap$(W_n) \le 3\varepsilon$, (4.1) now follows from (4.9), (4.11) and (4.13).

Having proven that $M_n \xrightarrow{\text{cap}} f$ in $D\setminus E$ at faster than geometric rate whenever it is a sequence of $n$-th root optimal meromorphic approximants to $f$, we turn to the construction of rational functions $R_{k_n} \in \mathcal{R}_{k_n}(D)$ such that the poles of $R_{k_n}$ are among the poles of $M_n$ lying in $V$ and (2.18) holds, where $V$ any open set such that $E \subset V \subset \bar{V} \subset D$. Let $B \supset E$ be a closed set contained in $V$ which is regular for the Dirichlet problem, see Section A.7. Such a $B$ is easily constructed as a sublevel set, for some small regular value, of a smooth non-negative function whose zero set is $E$, see, for example, discussion after (3.8). Then the Green equilibrium potential $G(z) := g(\mu_{D,B}, D; z)$ is harmonic in $D\setminus B$, continuous on $D$, strictly less than the constant $1/\text{cap}_D(B)$ on $D\setminus B$ and equal to that constant on $B$, see (2.12) or Section A.4 for a more detailed discussion. Since $\partial_z G(z)$ is holomorphic in $D\setminus B$, the critical points of $G$ are isolated and cannot accumulate in $D\setminus B$, so we can find an interval $[t_1, t_2] \subset (0, 1/\text{cap}_D(B))$ that is free of critical values and such that $G^{-1}([t_1, t_2]) \subset V$. For any $t \in [t_1, t_2]$, $\gamma(t) := G^{-1}(t)$ is a 1-dimensional compact manifold, i.e., a finite union of disjoint real analytic closed curves $\gamma_{1,t}, \dots, \gamma_{N,t}$, none of which lies interior to another (by the maximum principle), and such that $B \subset \text{int } \gamma(t) \subset \overline{\text{int } \gamma(t)} \subset V$. Note that $N$ is independent of $t \in [t_1, t_2]$ since any such $t$ is a regular value; note also that the total length $|\gamma(t)| = \sum_{j=1}^N |\gamma_{j,t}|$ is bounded above
where $|\gamma|$ stands for the arclength of $\gamma$. In another connection, it follows from [49, Lemma] that for any $\varepsilon \in (0, 1/3)$ there exists $W_n \subset \mathbb{D}$ such that $\mathrm{cap}(W_n) \le 3\varepsilon$ and

$$|p_k^\ell(\zeta)q_n(\zeta)| > \varepsilon^{n+\ell k} \|p_k^\ell q_n\|_{\mathbb{T}}, \quad \zeta \in \mathbb{D}\setminus W_n.$$

As $|\tilde{p}_k^\ell \tilde{q}_n|(z) \le \|\tilde{p}_k^\ell \tilde{q}_n\|_{\mathbb{T}} = \|p_k^\ell q_n\|_{\mathbb{T}}$ for $z \in \mathbb{D}$ by the maximum principle and the definition of the reciprocal polynomial, we get that $|(b_n B_k^\ell)(z)| \ge \varepsilon^{n+k\ell}$ for $z \in \mathbb{D}\setminus W_n$. Since $\mathbb{P}_+ + \mathbb{P}_-$ is the identity operator, we can use the analytic continuation provided by (4.7) to write

$$(4.9) \qquad e_n(z) = \frac{\mathbb{P}_+(e_n b_n B_k^\ell)(z) + \mathbb{P}_-(e_n b_n B_k^\ell)(z)}{b_n(z) B_k^\ell(z)}, \quad z \in \bar{\mathbb{D}}\setminus\overline{\mathrm{int} \, \gamma}.$$

The estimate $|(b_n B_k^\ell)(z)| \ge \varepsilon^{n+k\ell}$ for $z \in \mathbb{D}\setminus W_n$, (4.8), and the definition of $\ell$ give us that

$$(4.10) \qquad \left| \frac{\mathbb{P}_-(e_n b_n B_k^\ell)(z)}{b_n(z) B_k^\ell(z)} \right| \le \frac{1}{\varepsilon^{2n}} \left( \frac{\eta}{\mathrm{dist}(\mathbb{T}, E)} \right)^{n-k} \frac{|\gamma| \|f\|_{\gamma}}{\mathrm{dist}(z, \gamma)}, \quad z \in \bar{\mathbb{D}}\setminus(\overline{\mathrm{int} \, \gamma} \cup W_n).$$

Now, given $0 < \varepsilon < 1$ and $0 < a < 1$, choose $\eta$ in (4.2) so that $0 < \eta < a\varepsilon^2\mathrm{dist}(\mathbb{T}, E)$. Choice of $\eta$ of course fixes $k$ in (4.2). Then, since $K$ lies exterior to $\gamma$ because $K \cap L_\eta = \emptyset$, we get from (4.10) that there exists a natural number $n_0 = n_0(f, K, \varepsilon, \eta, \gamma)$ for which

$$(4.11) \qquad \left| \frac{\mathbb{P}_-(e_n b_n B_k^\ell)(z)}{b_n(z) B_k^\ell(z)} \right| < a^n, \quad n \ge n_0, \quad z \in K \setminus W_n.$$

Next, as $\|e_n b_n B_k^\ell\|_{\mathbb{T}} = \|e_n\|_{\mathbb{T}} \to 0$ faster than geometrically with $n$ by hypothesis, (4.8) and the triangle inequality yield that

$$(4.12) \qquad \|\mathbb{P}_+(e_n b_n B_k^\ell)\|_{\mathbb{T}} = \|e_n b_n B_k^\ell - \mathbb{P}_-(e_n b_n B_k^\ell)\|_{\mathbb{T}} \le C \left( \frac{\eta}{\mathrm{dist}(\mathbb{T}, E)} \right)^n, \quad n \ge n_0,$$

for some constant $C = C(f, E, \gamma)$ and some $n_0'$ depending on $C$ and the speed of approximation of $f$ by $h_n/b_n$. Subsequently, as in (4.11), we get from the estimate $|(b_n B_k^\ell)(z)| \ge \varepsilon^{n+k\ell}$ for $z \in \mathbb{D}\setminus W_n$, (4.12), and the maximum modulus principle that

$$(4.13) \qquad \left| \frac{\mathbb{P}_+(e_n b_n B_k^\ell)(z)}{b_n(z) B_k^\ell(z)} \right| < a^n, \quad n \ge n_0'', \quad z \in K \setminus W_n,$$

for some natural number $n_0'' = n_0''(f, K, \varepsilon, \eta, \gamma)$. Because $\varepsilon$ and $a$ can be arbitrarily small and cap($W_n$) $\le 3\varepsilon$, (4.1) now follows from (4.9), (4.11) and (4.13).

Having proven that $M_n \xrightarrow{\mathrm{cap}} f$ in $D\setminus E$ at faster than geometric rate whenever it is a sequence of $n$-th root optimal meromorphic approximants to $f$, we turn to the construction of rational functions $R_{k_n} \in \mathcal{R}_{k_n}(D)$ such that the poles of $R_{k_n}$ are among the poles of $M_n$ lying in $V$ and (2.18) holds, where $V$ any open set such that $E \subset V \subset \bar{V} \subset D$. Let $B \supset E$ be a closed set contained in $V$ which is regular for the Dirichlet problem, see Section A.7. Such a $B$ is easily constructed as a sublevel set, for some small regular value, of a smooth non-negative function whose zero set is $E$, see, for example, discussion after (3.8). Then the Green equilibrium potential $G(z) := g(\mu_{D,B}, D; z)$ is harmonic in $D\setminus B$, continuous on $D$, strictly less than the constant $1/\mathrm{cap}_D(B)$ on $D\setminus B$ and equal to that constant on $B$, see (2.12) or Section A.4 for a more detailed discussion. Since $\partial_z G(z)$ is holomorphic in $D\setminus B$, the critical points of $G$ are isolated and cannot accumulate in $D\setminus B$, so we can find an interval $[t_1, t_2] \subset (0, 1/\mathrm{cap}_D(B))$ that is free of critical values and such that $G^{-1}([t_1, t_2]) \subset V$. For any $t \in [t_1, t_2]$, $\gamma(t) := G^{-1}(t)$ is a 1-dimensional compact manifold, i.e., a finite union of disjoint real analytic closed curves $\gamma_{1,t}, \dots, \gamma_{N,t}$, none of which lies interior to another (by the maximum principle), and such that $B \subset \mathrm{int} \, \gamma(t) \subset \overline{\mathrm{int} \, \gamma(t)} \subset V$. Note that $N$ is independent of $t \in [t_1, t_2]$ since any such $t$ is a regular value; note also that the total length $|\gamma(t)| = \sum_{j=1}^N |\gamma_{j,t}|$ is bounded above
independently of $t$, say by a constant $L$, because the gradient $\nabla G$ is normal to $\gamma_{j,t}$ at its every point and therefore the divergence formula implies for any $t \in [t_1, t_2]$:

$$t|\gamma(t)| \min_{z \in G^{-1}(t)} \|\nabla G(z)\| \le t_2|\gamma(t_2)| \max_{z \in G^{-1}(t_2)} \|\nabla G(z)\| - \int_{G^{-1}([t,t_2])} \|\nabla G\|^2 dxdy.$$

Pick $a > 0$, set $K := \max\{\|\nabla G(z)\| : z \in G^{-1}([t_1, t_2])\}$ and let $n_a \in \mathbb{N}$ be so large that

$$(4.14) \quad \operatorname{cap}(\{z \in G^{-1}([t_1, t_2]) : |f(z) - M_n(z)| > a^n\}) < \frac{t_2 - t_1}{4K}, \quad n \ge n_a.$$

Such a $n_a$ exists by the first part of the proof. Let $A_n$ be the set whose capacity is estimated in (4.14). Assume for the moment that for each $t \in [t_1, t_2]$ there exists $z \in G^{-1}(t) \cap A_n$. Then the image of $A_n$ under $G/K : G^{-1}([t_1, t_2]) \to \mathbb{R}$ is equal to the interval $[t_1/K, t_2/K]$ whose capacity is $(t_2 - t_1)/(4K)$. However, since contractive maps do not increase the logarithmic capacity [53, Theorem 5.3.1], the capacity of $G(A_n)/K$ should be strictly smaller than $(t_2 - t_1)/(4K)$ by (4.14). Hence, for each $n \ge n_a$ there is $t_n \in [t_1, t_2]$ for which

$$(4.15) \quad \left| \int_{\gamma(t_n)} \frac{f(\xi) - M_n(\xi)}{z - \xi} \frac{d\xi}{2\pi i} \right| \le \frac{La^n}{2\pi \operatorname{dist}(T, \bar{V})}, \quad z \in T.$$

Pick a positive sequence $\{a_k\}$ converging to 0 and, without loss of generality, arrange things so that $n_{a_k} < n_{a_{k+1}}$. Define

$$J_n(z) := \int_{\gamma(t_{n_{a_k}})} \frac{M_n(\xi)}{z - \xi} \frac{d\xi}{2\pi i}, \quad z \in D \setminus \bar{V}, \quad n_{a_k} \le n < n_{a_{k+1}}.$$

Clearly, $J_n$ is a rational function retaining the singular part of $M_n$ inside the system of arcs $\gamma(t_{n_{a_k}})$, and it is of type $(k_n - 1, k_n)$ where $k_n \le n$ is the number of the poles of $M_n$ inside this system of arcs, counting multiplicities. If we put $R_{k_n} := J_n$, then since $a_k \to 0$ we get from (4.15) and the Cauchy formula that (2.18) holds, as desired.

## APPENDIX A. POTENTIAL THEORY ON A RIEMANN SURFACE

Even though the proof of Theorem 2.5 in Sections 3.3–3.10 was carried out for $D = \mathbb{D}$, this appendix is written for a general Jordan domain $D$ since specializing $D$ to the unit disk would only shorten the proofs of Lemmas A.5 and A.10 by a couple of paragraphs but otherwise would not lead to any further simplifications.

### A.1. Subharmonic Functions.
Let $d$ be the differential and $*$ the conjugation operators on a connected Riemann surface. The Laplacian $\Delta := d^*d$ takes smooth functions to 2-forms. If $U$ is an open subset of the surface and $u : U \to \mathbb{R}$ a locally integrable function (against the area-form $\mathbf{1}$, where $\mathbf{1}$ is the constant unit function, or equivalently against the Lebesgue measure $(i/2)dz \wedge d\bar{z}$ in any system of local coordinates $z, \bar{z}$), the distributional Laplacian $\Delta u$ is the 0-current acting on a smooth compactly supported functions $\varphi$ on $U$ by $\int u\Delta\varphi$. When $\Delta u = 0$, one says that $u$ is harmonic on $U$, and such functions are in fact smooth (even real analytic) by Weyl's lemma [18, Theorem 24.9]. Subharmonic functions on $U$ are defined as upper-semicontinuous functions $u : U \to [-\infty, \infty)$ such that, if $V$ is open in $U$ and $h : V \to \mathbb{R}$ is harmonic, then $u - h$ is either constant or fails to have a maximum in $V$. On open subsets of $\mathbb{C}$, this definition coincides with the usual one; see [53, Definition 2.2.1 & Theorem 2.4.1]. A superharmonic function is the negative of a subharmonic function. A difference of two subharmonic functions is sometimes called a $\delta$-subharmonic function.

Harmonicity and subharmonicity are local properties: $u$ is harmonic (resp. subharmonic) on $U$ if and only if its restriction to every open subset is, or equivalently if and only if $u \circ \varphi^{-1}$ is harmonic (resp. subharmonic) on the open set $\varphi(V \cap U) \subset \mathbb{C}$ whenever $(V, \varphi)$ is a local chart. Thus, standard facts regarding such functions on open subsets of a Riemann surface follow from their planar counterparts, using charts. In particular, the integrability theorem [53, Theorem 2.5.1] states
independently of $t$, say by a constant $L$, because the gradient $\nabla G$ is normal to $\gamma_{j,t}$ at its every point and therefore the divergence formula implies for any $t \in [t_1, t_2]$:

$$t|\gamma(t)| \min_{z \in G^{-1}(t)} \|\nabla G(z)\| \le t_2|\gamma(t_2)| \max_{z \in G^{-1}(t_2)} \|\nabla G(z)\| - \int_{G^{-1}([t,t_2])} \|\nabla G\|^2 dxdy.$$

Pick $a > 0$, set $K := \max\{\|\nabla G(z)\| : z \in G^{-1}([t_1, t_2])\}$ and let $n_a \in \mathbb{N}$ be so large that

$$(4.14) \quad \operatorname{cap}(\{z \in G^{-1}([t_1, t_2]) : |f(z) - M_n(z)| > a^n\}) < \frac{t_2 - t_1}{4K}, \quad n \ge n_a.$$

Such a $n_a$ exists by the first part of the proof. Let $A_n$ be the set whose capacity is estimated in (4.14). Assume for the moment that for each $t \in [t_1, t_2]$ there exists $z \in G^{-1}(t) \cap A_n$. Then the image of $A_n$ under $G/K : G^{-1}([t_1, t_2]) \to \mathbb{R}$ is equal to the interval $[t_1/K, t_2/K]$ whose capacity is $(t_2 - t_1)/(4K)$. However, since contractive maps do not increase the logarithmic capacity [53, Theorem 5.3.1], the capacity of $G(A_n)/K$ should be strictly smaller than $(t_2 - t_1)/(4K)$ by (4.14). Hence, for each $n \ge n_a$ there is $t_n \in [t_1, t_2]$ for which

$$(4.15) \quad \left| \int_{\gamma(t_n)} \frac{f(\xi) - M_n(\xi)}{z - \xi} \frac{d\xi}{2\pi i} \right| \le \frac{La^n}{2\pi \operatorname{dist}(T, \bar{V})}, \quad z \in T.$$

Pick a positive sequence $\{a_k\}$ converging to 0 and, without loss of generality, arrange things so that $n_{a_k} < n_{a_{k+1}}$. Define

$$J_n(z) := \int_{\gamma(t_{n_{a_k}})} \frac{M_n(\xi)}{z - \xi} \frac{d\xi}{2\pi i}, \quad z \in D \setminus \bar{V}, \quad n_{a_k} \le n < n_{a_{k+1}}.$$

Clearly, $J_n$ is a rational function retaining the singular part of $M_n$ inside the system of arcs $\gamma(t_{n_{a_k}})$, and it is of type $(k_n - 1, k_n)$ where $k_n \le n$ is the number of the poles of $M_n$ inside this system of arcs, counting multiplicities. If we put $R_{k_n} := J_n$, then since $a_k \to 0$ we get from (4.15) and the Cauchy formula that (2.18) holds, as desired.

## APPENDIX A. POTENTIAL THEORY ON A RIEMANN SURFACE

Even though the proof of Theorem 2.5 in Sections 3.3–3.10 was carried out for $D = \mathbb{D}$, this appendix is written for a general Jordan domain $D$ since specializing $D$ to the unit disk would only shorten the proofs of Lemmas A.5 and A.10 by a couple of paragraphs but otherwise would not lead to any further simplifications.

### A.1. Subharmonic Functions.
Let $d$ be the differential and $*$ the conjugation operators on a connected Riemann surface. The Laplacian $\Delta := d^*d$ takes smooth functions to 2-forms. If $U$ is an open subset of the surface and $u : U \to \mathbb{R}$ a locally integrable function (against the area-form $\mathbf{1}$, where $\mathbf{1}$ is the constant unit function, or equivalently against the Lebesgue measure $(i/2)dz \wedge d\bar{z}$ in any system of local coordinates $z, \bar{z}$), the distributional Laplacian $\Delta u$ is the 0-current acting on a smooth compactly supported functions $\varphi$ on $U$ by $\int u\Delta\varphi$. When $\Delta u = 0$, one says that $u$ is harmonic on $U$, and such functions are in fact smooth (even real analytic) by Weyl's lemma [18, Theorem 24.9]. Subharmonic functions on $U$ are defined as upper-semicontinuous functions $u : U \to [-\infty, \infty)$ such that, if $V$ is open in $U$ and $h : V \to \mathbb{R}$ is harmonic, then $u - h$ is either constant or fails to have a maximum in $V$. On open subsets of $\mathbb{C}$, this definition coincides with the usual one; see [53, Definition 2.2.1 & Theorem 2.4.1]. A superharmonic function is the negative of a subharmonic function. A difference of two subharmonic functions is sometimes called a $\delta$-subharmonic function.

Harmonicity and subharmonicity are local properties: $u$ is harmonic (resp. subharmonic) on $U$ if and only if its restriction to every open subset is, or equivalently if and only if $u \circ \varphi^{-1}$ is harmonic (resp. subharmonic) on the open set $\varphi(V \cap U) \subset \mathbb{C}$ whenever $(V, \varphi)$ is a local chart. Thus, standard facts regarding such functions on open subsets of a Riemann surface follow from their planar counterparts, using charts. In particular, the integrability theorem [53, Theorem 2.5.1] states
that a subharmonic function which is not identically $-\infty$ is locally integrable, and therefore it has a distributional Laplacian. Hence, two subharmonic functions that coincide almost everywhere (with respect to area measure) are in fact equal, for either they are both identically $-\infty$ or they have the same distributional Laplacian, and so their difference is harmonic; this is the weak identity principle. The following is a variant of Harnack's theorem [53, Theorem 1.3.10] and of [53, Theorem 2.4.6].

**Harnack's Theorem.** A sequence of harmonic functions on $U$ that is bounded below has a sub-sequence that converges locally uniformly on $U$, either to $+\infty$ or to a harmonic function. For an increasing sequence, convergence holds along the full sequence. A decreasing sequence of subharmonic functions converges pointwise to a subharmonic function.

A locally integrable function $u$ is subharmonic if and only if $\Delta u$ is a Radon measure on the surface; that is, $\Delta u$ is a positive linear form on continuous functions with compact support. Indeed, as this statement is local, it reduces to its planar analog. The "only if" part follows from [53, Section 3.7]. As to the "if" part, let $W \subset \mathbb{C}$ be open and $v$ be a finite positive Borel measure carried by $W$. The logarithmic potential of $v$, i.e., $V^v(z) := \int \log|z-t|^{-1}dv(t)$, is superharmonic on $\mathbb{C}$ with distributional Laplacian $-v$, so if $u$ is locally integrable on $W$ with $\Delta u = v$ there, then $h := u + V^v_{[W}]$ is harmonic and therefore $u = -V^v_{[W]} + h$ is subharmonic in $W$ (as in the main text, for a set $E$ (that may require further qualification), a subscript $\lfloor E$ indicates "restriction to $E$").

### A.2. Green Functions.
Throughout, $\Omega$ will be a subdomain of some ambient algebraic Riemann surface $\mathcal{R}_*$ such that $p(\Omega)$ is a bounded domain in $\mathbb{C}$, where $p$ stands for the canonical projection; in particular the results apply to $\Omega = \mathcal{R}$ with $p(\Omega) = D$, see beginning of Section 2.1. Since the lift to $\Omega$ of a positive non-constant superharmonic function on $p(\Omega)$ is again positive, non-constant, and superharmonic, $\Omega$ is hyperbolic and as such possesses Green functions [15, Theorem IV.3.7]. Notice also that there exists a subdomain $\Omega' \subset \mathcal{R}_*$ with $p(\Omega')$ bounded such that $\bar{\Omega} \subset \Omega'$, where an overline (as in $\bar{\Omega}$) always denotes the closure in $\mathcal{R}_*$.

Recall that the Green function for $\Omega$ with pole at $w$, denoted by $g_{\Omega}(\cdot, w)$, is the unique function that is harmonic and positive in $\Omega \setminus \{w\}$ with a logarithmic singularity at $w$ and whose largest harmonic minorant is identically zero. By a logarithmic singularity at $w$, it is meant that if $(V, \varphi)$ is a coordinate chart on $\Omega$ such that $w \in V$ and $\varphi(w) = 0$, then $g_{\Omega}(\varphi^{-1}(\cdot), w) + \log|\cdot|$ is harmonic on $\varphi(V)$. Obviously $g_{\Omega}(\cdot, w) > 0$ everywhere on $\Omega$, by the minimum principle for harmonic functions. Recall also that if a superharmonic function on $\Omega$ is not identically $+\infty$ and has a harmonic minorant, then it has the largest one whose construction can be carried out as in the Euclidean case [29, Theorem 4.3.5], because Poisson modifications can be performed locally.

Clearly, $g_{\Omega}(\cdot, w)$ is superharmonic and $\Delta g_{\Omega}(\cdot, w) = -\delta_w$, where $\delta_w$ is the Dirac mass at $w$. Moreover, $g_{\Omega}$ is symmetric in that $g_{\Omega}(z, w) = g_{\Omega}(w, z)$ [15, Theorem IV.3.10]. Symmetry entails that $g_{\Omega}(z, w)$ is separately harmonic in $z$ and $w$, and therefore jointly harmonic on $\{(z, w) \in \Omega \times \Omega : z \neq w\}$ [36, p. 561]; in particular, $(z, w) \mapsto g_{\Omega}(z, w)$ is continuous off the diagonal. Note that

$$ (A.1) \qquad g_{\Omega}(z, w) \le g_{\Omega'}(z, w), \quad z, w \in \Omega \subseteq \Omega', $$

because $g_{\Omega}(\cdot, w) - g_{\Omega'}(\cdot, w)$ is a harmonic minorant of $g_{\Omega}(\cdot, w)$ and therefore must be non-positive. Thus, if $F_1, F_2$ are relatively closed subsets of $\Omega$ with $\bar{F}_1 \cap \bar{F}_2 = \emptyset$, we deduce that $(z, w) \mapsto g_{\Omega}(z, w)$ is bounded on $F_1 \times F_2$ because $g_{\Omega'}(z, w)$ is continuous on the compact set $\bar{F}_1 \times \bar{F}_2$ whenever $\bar{\Omega} \subset \Omega'$. We also remark that to each $w \in \Omega$ there is an open set $V \ni w$ and a constant $C = C(V)$ such that

$$ (A.2) \qquad \int_V g_{\Omega}(z, w') * \mathbf{1}(z) < C, \quad w' \in V, $$

a result that follows by uniformization from the corresponding fact on the disk [29, Theorem 4.4.12]. Moreover, if a sequence of open sets $\Omega_n$ increases to $\Omega$ as $n \to \infty$, it follows from (A.1) that $g_{\Omega}(\cdot, w)_{\lfloor \Omega_n} - g_{\Omega_n}(\cdot, w)$ is a decreasing sequence of positive harmonic functions that must converge
that a subharmonic function which is not identically $-\infty$ is locally integrable, and therefore it has a distributional Laplacian. Hence, two subharmonic functions that coincide almost everywhere (with respect to area measure) are in fact equal, for either they are both identically $-\infty$ or they have the same distributional Laplacian, and so their difference is harmonic; this is the weak identity principle. The following is a variant of Harnack's theorem [53, Theorem 1.3.10] and of [53, Theorem 2.4.6].

**Harnack's Theorem.** A sequence of harmonic functions on $U$ that is bounded below has a sub-sequence that converges locally uniformly on $U$, either to $+\infty$ or to a harmonic function. For an increasing sequence, convergence holds along the full sequence. A decreasing sequence of subharmonic functions converges pointwise to a subharmonic function.

A locally integrable function $u$ is subharmonic if and only if $\Delta u$ is a Radon measure on the surface; that is, $\Delta u$ is a positive linear form on continuous functions with compact support. Indeed, as this statement is local, it reduces to its planar analog. The "only if" part follows from [53, Section 3.7]. As to the "if" part, let $W \subset \mathbb{C}$ be open and $v$ be a finite positive Borel measure carried by $W$. The logarithmic potential of $v$, i.e., $V^v(z) := \int \log|z-t|^{-1}dv(t)$, is superharmonic on $\mathbb{C}$ with distributional Laplacian $-v$, so if $u$ is locally integrable on $W$ with $\Delta u = v$ there, then $h := u + V^v_{[W}]$ is harmonic and therefore $u = -V^v_{[W]} + h$ is subharmonic in $W$ (as in the main text, for a set $E$ (that may require further qualification), a subscript $\ll E$ indicates "restriction to $E$”).

**A.2. Green Functions.** Throughout, $\Omega$ will be a subdomain of some ambient algebraic Riemann surface $\mathcal{R}_*$ such that $p(\Omega)$ is a bounded domain in $\mathbb{C}$, where $p$ stands for the canonical projection; in particular the results apply to $\Omega = \mathcal{R}$ with $p(\Omega) = D$, see beginning of Section 2.1. Since the lift to $\Omega$ of a positive non-constant superharmonic function on $p(\Omega)$ is again positive, non-constant, and superharmonic, $\Omega$ is hyperbolic and as such possesses Green functions [15, Theorem IV.3.7]. Notice also that there exists a subdomain $\Omega' \subset \mathcal{R}_*$ with $p(\Omega')$ bounded such that $\bar{\Omega} \subset \Omega'$, where an overline (as in $\bar{\Omega}$) always denotes the closure in $\mathcal{R}_*$.

Recall that the Green function for $\Omega$ with pole at $w$, denoted by $g_\Omega(\cdot, w)$, is the unique function that is harmonic and positive in $\Omega \setminus \{w\}$ with a logarithmic singularity at $w$ and whose largest harmonic minorant is identically zero. By a logarithmic singularity at $w$, it is meant that if $(V, \varphi)$ is a coordinate chart on $\Omega$ such that $w \in V$ and $\varphi(w) = 0$, then $g_\Omega(\varphi^{-1}(\cdot), w) + \log|\cdot|$ is harmonic on $\varphi(V)$. Obviously $g_\Omega(\cdot, w) > 0$ everywhere on $\Omega$, by the minimum principle for harmonic functions. Recall also that if a superharmonic function on $\Omega$ is not identically $+\infty$ and has a harmonic minorant, then it has the largest one whose construction can be carried out as in the Euclidean case [29, Theorem 4.3.5], because Poisson modifications can be performed locally.

Clearly, $g_\Omega(\cdot, w)$ is superharmonic and $\Delta g_\Omega(\cdot, w) = -\delta_w$, where $\delta_w$ is the Dirac mass at $w$. Moreover, $g_\Omega$ is symmetric in that $g_\Omega(z, w) = g_\Omega(w, z)$ [15, Theorem IV.3.10]. Symmetry entails that $g_\Omega(z, w)$ is separately harmonic in $z$ and $w$, and therefore jointly harmonic on $\{(z, w) \in \Omega \times \Omega : z \neq w\}$ [36, p. 561]; in particular, $(z, w) \mapsto g_\Omega(z, w)$ is continuous off the diagonal. Note that

$$ (A.1) \qquad g_{\Omega}(z, w) \le g'_{\Omega}(z, w), \quad z, w \in \Omega \subseteq \Omega', $$

because $g_\Omega(\cdot, w) - g'_{\Omega}(\cdot, w)$ is a harmonic minorant of $g_\Omega(\cdot, w)$ and therefore must be non-positive. Thus, if $F_1, F_2$ are relatively closed subsets of $\Omega$ with $\bar{F}_1 \cap \bar{F}_2 = \emptyset$, we deduce that $(z, w) \mapsto g_\Omega(z, w)$ is bounded on $F_1 \times F_2$ because $g'_{\Omega}(z, w)$ is continuous on the compact set $\bar{F}_1 \times \bar{F}_2$ whenever $\bar{\Omega} \subset \Omega'$. We also remark that to each $w \in \Omega$ there is an open set $V \ni w$ and a constant $C = C(V)$ such that

$$ (A.2) \qquad \int_V g_\Omega(z, w') * \mathbf{1}(z) < C, \quad w' \in V, $$

a result that follows by uniformization from the corresponding fact on the disk [29, Theorem 4.4.12]. Moreover, if a sequence of open sets $\Omega_n$ increases to $\Omega$ as $n \to \infty$, it follows from (A.1) that $g_\Omega(\cdot, w)_{[\Omega_n]} - g_{\Omega_n}(\cdot, w)$ is a decreasing sequence of positive harmonic functions that must converge