# n-th Root Optimal Rational Approximants to Functions with Polar Singular Set

Laurent Baratchart, Herbert Stahl, Maxim Yattselev

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Laurent Baratchart, Herbert Stahl, Maxim Yattselev. n-th Root Optimal Rational Approximants to Functions with Polar Singular Set. 2024. hal-04604029

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# N-TH ROOT OPTIMAL RATIONAL APPROXIMANTS TO FUNCTIONS WITH POLAR SINGULAR SET

LAURENT BARATCHART, HERBERT STAHL†, AND MAXIM YATTSELEV

**ABSTRACT.** Let $D$ be a bounded Jordan domain and $A$ be its complement on the Riemann sphere. We investigate the $n$-th root asymptotic behavior in $D$ of best rational approximants, in the uniform norm on $A$, to functions holomorphic on $A$ having a multi-valued continuation to quasi every point of $D$ with finitely many branches. More precisely, we study weak* convergence of the normalized counting measures of the poles of such approximants as well as their convergence in capacity. We place best rational approximants into a larger class of $n$-th root optimal meromorphic approximants, whose behavior we investigate using potential-theory on certain compact bordered Riemann surfaces.

## CONTENTS

<table><tr><td>List of Symbols</td><td>2</td></tr><tr><td>1 Introduction</td><td>3</td></tr><tr><td>2 Preliminaries and Main Results</td><td>5</td></tr><tr><td>2.1 Class of Approximated Functions</td><td>6</td></tr><tr><td>2.2 Optimal Rational Approximants</td><td>8</td></tr><tr><td>2.3 Optimal Meromorphic Approximants</td><td>11</td></tr><tr><td>3 Proof of Theorem 2.5</td><td>13</td></tr><tr><td>3.1 Existence of Best Meromorphic Approximants</td><td>13</td></tr><tr><td>3.2 Reduction to the Unit Disk</td><td>13</td></tr><tr><td>3.3 Nehari Modifications</td><td>14</td></tr><tr><td>3.4 Notation</td><td>15</td></tr><tr><td>3.5 Stripped Error of Approximation</td><td>16</td></tr><tr><td>3.6 Asymptotic Distributions of Poles and Zeros</td><td>20</td></tr><tr><td>3.7 Logarithmic Error Function</td><td>23</td></tr><tr><td>3.8 Modified Logarithmic Error Function</td><td>29</td></tr><tr><td>3.9 Projected Logarithmic Error Function</td><td>33</td></tr><tr><td>3.10 Computation of the Logarithmic Error Function</td><td>35</td></tr><tr><td>3.11 Convergence in Capacity</td><td>37</td></tr><tr><td>4 Proof of Theorem 2.6</td><td>39</td></tr><tr><td>Appendix A Potential Theory on a Riemann Surface</td><td>42</td></tr><tr><td>A.1 Subharmonic Functions</td><td>42</td></tr><tr><td>A.2 Green Functions</td><td>43</td></tr><tr><td>A.3 Green Potentials</td><td>44</td></tr><tr><td>A.4 Capacities</td><td>45</td></tr><tr><td>A.5 Fine Topology</td><td>46</td></tr><tr><td>A.6 Thinness</td><td>47</td></tr></table>

<footnote n="2020">2020 Mathematics Subject Classification. 30C15, 30E10, 31C40, 41A20, 41A25.</footnote>

<footnote>**Key words and phrases.** Rational approximation, meromorphic approximation, AAK approximation, Nehari approximation, weak distribution of poles, convergence in capacity.</footnote>

<footnote>The research of the last author was supported in part by a grant from the Simons Foundation, CGM-706591.</footnote>
# N-TH ROOT OPTIMAL RATIONAL APPROXIMANTS TO FUNCTIONS WITH POLAR SINGULAR SET

LAURENT BARATCHART, HERBERT STAHL†, AND MAXIM YATTSELEV

**ABSTRACT.** Let $D$ be a bounded Jordan domain and $A$ be its complement on the Riemann sphere. We investigate the $n$-th root asymptotic behavior in $D$ of best rational approximants, in the uniform norm on $A$, to functions holomorphic on $A$ having a multi-valued continuation to quasi every point of $D$ with finitely many branches. More precisely, we study weak* convergence of the normalized counting measures of the poles of such approximants as well as their convergence in capacity. We place best rational approximants into a larger class of $n$-th root optimal meromorphic approximants, whose behavior we investigate using potential-theory on certain compact bordered Riemann surfaces.

## CONTENTS

<table><tr><td>List of Symbols</td><td>2</td></tr><tr><td>1 Introduction</td><td>3</td></tr><tr><td>2 Preliminaries and Main Results</td><td>5</td></tr><tr><td>2.1 Class of Approximated Functions</td><td>6</td></tr><tr><td>2.2 Optimal Rational Approximants</td><td>8</td></tr><tr><td>2.3 Optimal Meromorphic Approximants</td><td>11</td></tr><tr><td>3 Proof of Theorem 2.5</td><td>13</td></tr><tr><td>3.1 Existence of Best Meromorphic Approximants</td><td>13</td></tr><tr><td>3.2 Reduction to the Unit Disk</td><td>13</td></tr><tr><td>3.3 Nehari Modifications</td><td>14</td></tr><tr><td>3.4 Notation</td><td>15</td></tr><tr><td>3.5 Stripped Error of Approximation</td><td>16</td></tr><tr><td>3.6 Asymptotic Distributions of Poles and Zeros</td><td>20</td></tr><tr><td>3.7 Logarithmic Error Function</td><td>23</td></tr><tr><td>3.8 Modified Logarithmic Error Function</td><td>29</td></tr><tr><td>3.9 Projected Logarithmic Error Function</td><td>33</td></tr><tr><td>3.10 Computation of the Logarithmic Error Function</td><td>35</td></tr><tr><td>3.11 Convergence in Capacity</td><td>37</td></tr><tr><td>4 Proof of Theorem 2.6</td><td>39</td></tr><tr><td>Appendix A Potential Theory on a Riemann Surface</td><td>42</td></tr><tr><td>A.1 Subharmonic Functions</td><td>42</td></tr><tr><td>A.2 Green Functions</td><td>43</td></tr><tr><td>A.3 Green Potentials</td><td>44</td></tr><tr><td>A.4 Capacities</td><td>45</td></tr><tr><td>A.5 Fine Topology</td><td>46</td></tr><tr><td>A.6 Thinness</td><td>47</td></tr></table>

<footnote n="2020">2020 Mathematics Subject Classification. 30C15, 30E10, 31C40, 41A20, 41A25.</footnote>

<footnote>**Key words and phrases.** Rational approximation, meromorphic approximation, AAK approximation, Nehari approximation, weak distribution of poles, convergence in capacity.</footnote>

<footnote>The research of the last author was supported in part by a grant from the Simons Foundation, CGM-706591.</footnote>
<table><tr><td>A.7</td><td>Regularity</td><td>49</td></tr><tr><td>A.8</td><td>Superlevel Sets of Green Potentials</td><td>51</td></tr><tr><td>A.9</td><td>Balayage</td><td>53</td></tr><tr><td>A.10</td><td>Green Potentials in D and on R</td><td>55</td></tr><tr><td>A.11</td><td>Dirichlet problem</td><td>57</td></tr><tr><td colspan="2">References</td><td>59</td></tr></table>

## LIST OF SYMBOLS

### General Point Sets:

<table><tr><td>T, D, C, $\bar{C}$</td><td>unit circle, open unit disk, complex plane, and extended complex plane</td></tr><tr><td>T<sub>r</sub>, D<sub>r</sub></td><td>circle and open disk of radius r centered at the origin</td></tr><tr><td>T, D, A</td><td>Jordan curve, its interior domain, and the closure of its exterior domain in $\bar{C}$</td></tr></table>

### Riemann Surfaces:

<table><tr><td>$\mathcal{R}_*, p$</td><td>compact Riemann surface with natural projection $p: \mathcal{R}_* \to \bar{C}$</td></tr><tr><td>$\mathcal{R}$</td><td>$\mathcal{R} := \{z \in \mathcal{R}_* : p(z) \in D\}$</td></tr><tr><td>$\mathcal{T}$</td><td>a connected component of $p^{-1}(T)$ homeomorphic to $T$</td></tr><tr><td>$\mathbf{rp}(\cdot)$</td><td>the set of ramification points of a given Riemann surface</td></tr><tr><td>$m(z)$</td><td>ramification order of a point z on a Riemann surface</td></tr><tr><td>$M$</td><td>total number of sheets of $\mathcal{R}$</td></tr></table>

### Classes of Functions:

<table><tr><td>$C(E)$</td><td>continuous functions on a set E</td></tr><tr><td>$\mathcal{H}(A)$</td><td>functions analytic in some neighborhood of A</td></tr><tr><td>$S(A)$</td><td>subclass of $\mathcal{H}(A)$ of functions multi-valued and quasi everywhere analytic off A</td></tr><tr><td>$\mathcal{E}(A)$</td><td>subclass of $\mathcal{H}(A)$ of functions single-valued and quasi everywhere analytic off A</td></tr><tr><td>$\mathcal{F}(\mathcal{R})$</td><td>class of functions quasi everywhere analytic on $\bar{\mathcal{R}}$</td></tr><tr><td>$\mathcal{F}(A)$</td><td>subclass of approximated functions analytic on A</td></tr><tr><td>$\mathcal{P}_n$</td><td>algebraic polynomials of degree at most n</td></tr><tr><td>$\mathcal{M}_n(D)$</td><td>monic algebraic polynomials of degree n with all their zeros in D</td></tr><tr><td>$\mathcal{R}_n(D)$</td><td>$\mathcal{P}_n \mathcal{M}_n^{-1}(D)$</td></tr><tr><td>$H^\infty(D)$</td><td>space of bounded holomorphic functions in D</td></tr><tr><td>$H_n^\infty(D)$</td><td>$H^\infty(D) \mathcal{M}_n^{-1}(D)$</td></tr><tr><td>$L^2(\mathbb{T})$</td><td>square integrable functions on the unit circle</td></tr><tr><td>$L^\infty(\mathbb{T})$</td><td>essentially bounded functions on the unit circle</td></tr><tr><td>$H^2$</td><td>Hardy space of functions in $L^2(\mathbb{T})$ with vanishing Fourier coefficients of negative index</td></tr><tr><td>$H^2_-$</td><td>$L^2(\mathbb{T}) \ominus H^2$</td></tr></table>

### Operators:

<table><tr><td>$\mathbb{P}_+, \mathbb{P}_-$</td><td>orthogonal projections from $L^2(\mathbb{T})$ onto $H^2, H^2_-$</td></tr><tr><td>$\Gamma_f$</td><td>Hankel operator from $H^2$ to $H^2_-$, $h \mapsto \mathbb{P}_-(fh)$</td></tr><tr><td>$s_n(\Gamma_f)$</td><td>the n-th singular number of $\Gamma_f$</td></tr></table>

### Potential Theory:
<table><tr><td>A.7</td><td>Regularity</td><td>49</td></tr><tr><td>A.8</td><td>Superlevel Sets of Green Potentials</td><td>51</td></tr><tr><td>A.9</td><td>Balayage</td><td>53</td></tr><tr><td>A.10</td><td>Green Potentials in D and on R</td><td>55</td></tr><tr><td>A.11</td><td>Dirichlet problem</td><td>57</td></tr><tr><td colspan="2">References</td><td>59</td></tr></table>

## LIST OF SYMBOLS

### General Point Sets:

<table><tr><td>T, D, C, $\bar{C}$</td><td>unit circle, open unit disk, complex plane, and extended complex plane</td></tr><tr><td>T<sub>r</sub>, D<sub>r</sub></td><td>circle and open disk of radius r centered at the origin</td></tr><tr><td>T, D, A</td><td>Jordan curve, its interior domain, and the closure of its exterior domain in $\bar{C}$</td></tr></table>

### Riemann Surfaces:

<table><tr><td>$\mathcal{R}_*, p$</td><td>compact Riemann surface with natural projection $p: \mathcal{R}_* \to \bar{C}$</td></tr><tr><td>$\mathcal{R}$</td><td>$\mathcal{R} := \{z \in \mathcal{R}_* : p(z) \in D\}$</td></tr><tr><td>$\mathcal{T}$</td><td>a connected component of $p^{-1}(T)$ homeomorphic to $T$</td></tr><tr><td>$\mathbf{rp}(\cdot)$</td><td>the set of ramification points of a given Riemann surface</td></tr><tr><td>$m(z)$</td><td>ramification order of a point z on a Riemann surface</td></tr><tr><td>$M$</td><td>total number of sheets of $\mathcal{R}$</td></tr></table>

### Classes of Functions:

<table><tr><td>$C(E)$</td><td>continuous functions on a set E</td></tr><tr><td>$\mathcal{H}(A)$</td><td>functions analytic in some neighborhood of A</td></tr><tr><td>$S(A)$</td><td>subclass of $\mathcal{H}(A)$ of functions multi-valued and quasi everywhere analytic off A</td></tr><tr><td>$\mathcal{E}(A)$</td><td>subclass of $\mathcal{H}(A)$ of functions single-valued and quasi everywhere analytic off A</td></tr><tr><td>$\mathcal{F}(\mathcal{R})$</td><td>class of functions quasi everywhere analytic on $\bar{\mathcal{R}}$</td></tr><tr><td>$\mathcal{F}(A)$</td><td>subclass of approximated functions analytic on A</td></tr><tr><td>$\mathcal{P}_n$</td><td>algebraic polynomials of degree at most n</td></tr><tr><td>$\mathcal{M}_n(D)$</td><td>monic algebraic polynomials of degree n with all their zeros in D</td></tr><tr><td>$\mathcal{R}_n(D)$</td><td>$\mathcal{P}_n \mathcal{M}_n^{-1}(D)$</td></tr><tr><td>$H^\infty(D)$</td><td>space of bounded holomorphic functions in D</td></tr><tr><td>$H_n^\infty(D)$</td><td>$H^\infty(D) \mathcal{M}_n^{-1}(D)$</td></tr><tr><td>$L^2(\mathbb{T})$</td><td>square integrable functions on the unit circle</td></tr><tr><td>$L^\infty(\mathbb{T})$</td><td>essentially bounded functions on the unit circle</td></tr><tr><td>$H^2$</td><td>Hardy space of functions in $L^2(\mathbb{T})$ with vanishing Fourier coefficients of negative index</td></tr><tr><td>$H^2_-$</td><td>$L^2(\mathbb{T}) \ominus H^2$</td></tr></table>

### Operators:

<table><tr><td>$\mathbb{P}_+, \mathbb{P}_-$</td><td>orthogonal projections from $L^2(\mathbb{T})$ onto $H^2, H^2_-$</td></tr><tr><td>$\Gamma_f$</td><td>Hankel operator from $H^2$ to $H^2_-$, $h \mapsto \mathbb{P}_-(fh)$</td></tr><tr><td>$s_n(\Gamma_f)$</td><td>the n-th singular number of $\Gamma_f$</td></tr></table>

### Potential Theory:
cap(K) logarithmic capacity of K

cap<sub>Ω</sub>(K) Greenian capacity of K relative to Ω

cap(E, K) capacity of the condenser (E, K)

g<sub>Ω</sub>(·, w) Green function for Ω with pole at w

g(σ, Ω; z) Green potential of the measure σ relative to Ω

V<sup>σ</sup>(z) logarithmic potential of the measure σ

μ<sub>Ω,K</sub> Green equilibrium distribution on a set K ⊂ Ω relative to Ω

B<sub>v</sub><sup>E</sup> balayage function of a superharmonic function v relative to a set E

σ<sup>E</sup> balayage measure of a measure σ onto a set E

σ̂ lift of a measure σ on D to R

p<sub>*</sub>(σ) projection (pushforward) of a measure σ on R to D

∂<sub>f</sub>A, clos<sub>f</sub>(E) fine boundary and closure of a set E

b(E), i(E) base and the set of finely isolated points of a set E

Various Symbols:

φ a conformal map from D onto D

K<sub>f</sub> collection of “branch cuts” for f

K<sub>f</sub> “branch cut” of minimal Greenian capacity for f ∈ S(A)

‖·‖<sub>K</sub> essential supremum norm on K

ρ<sub>n</sub>(f, A) error of best rational approximation of f analytic on A by functions in R<sub>n</sub>(D)

# 1. INTRODUCTION

Rational approximation to holomorphic functions of one complex variable has long been a requisite chapter of classical analysis with notable applications to number theory [30, 56, 33], spectral theory [44, 26, 48] and numerical analysis [25, 32, 24, 17]. Over the last decades it became a cornerstone of modeling in Engineering [65, 66, 51, 1, 27], and it can also be viewed today as a technique to regularize inverse potential problems [22, 31, 4]. Finding best rational approximants of prescribed degree to a specific function, say in the uniform norm on a given set, seems out of reach except in rare, particular cases. Indeed, such approximants depend in a rather convoluted manner, both on the approximated function and on the set where approximation should take place. Accordingly, the constructive side of the theory has focused on estimating optimal convergence rates as the degree grows large and devising approximation schemes coming close to meet them [67, 7, 23, 62], or else studying the behavior of natural, computationally appealing candidate approximants like Padé interpolants and their variants [3, 61, 45, 38, 42].

When a function is holomorphic in some neighborhood of a continuum A, the optimal speed of convergence for rational approximants on A is at least geometric in the degree. Then, a coarse but manageable estimate of this speed proceeds via asymptotics of the n-th root of the error in approximation by rational functions of degree n. For functions analytically continuable off A except over a polar set containing branchpoints (throughout *polar* means of logarithmic capacity zero), and provided that A does not divide the extended complex plane, Gonchar and Rakhmanov constructed, using multipoint Padé interpolants and dwelling on work by the second author, a sequence of rational approximants whose n-th root error on A is asymptotically the smallest possible. They further showed that these interpolants converge in capacity on the complement of a compact set K minimizing the capacity of the condenser (A, K) under the constraint that the function is analytic off K, and proved that the normalized counting measures of their poles converge to the condenser equilibrium distribution on K [23]. It is remarkable that K solves a geometric extremal problem from logarithmic potential theory, close in spirit to the Lavrentiev type [34], and that it depends merely on the set where approximation takes place, on the branchpoints of the approximated function and its monodromy, but nothing else. Such a structure emerges because only the n-th root of the error is considered, rather than the error itself. Since then, it has been an open issue whether any n-th root
$cap(K)$ logarithmic capacity of $K$

$cap_Ω(K)$ Greenian capacity of $K$ relative to $Ω$

$cap(E, K)$ capacity of the condenser $(E, K)$

$g_Ω(\cdot, w)$ Green function for $Ω$ with pole at $w$

$g(σ, Ω; z)$ Green potential of the measure $σ$ relative to $Ω$

$V^σ(z)$ logarithmic potential of the measure $σ$

$μ_{Ω,K}$ Green equilibrium distribution on a set $K ⊂ Ω$ relative to $Ω$

$B_ν^E$ balayage function of a superharmonic function $ν$ relative to a set $E$

$σ^E$ balayage measure of a measure $σ$ onto a set $E$

$\hat{\sigma}$ lift of a measure $σ$ on $D$ to $R$

$p_*(σ)$ projection (pushforward) of a measure $σ$ on $R$ to $D$

$∂_fA, \text{ clos}_f(E)$ fine boundary and closure of a set $E$

$b(E), i(E)$ base and the set of finely isolated points of a set $E$

**Various Symbols:**

$\phi$ a conformal map from $D$ onto $D$

$K_f$ collection of “branch cuts” for $f$

$K_f$ “branch cut” of minimal Greenian capacity for $f \in S(A)$

$|| \cdot ||_K$ essential supremum norm on $K$

$ρ_n(f, A)$ error of best rational approximation of $f$ analytic on $A$ by functions in $R_n(D)$

# 1. INTRODUCTION

Rational approximation to holomorphic functions of one complex variable has long been a requisite chapter of classical analysis with notable applications to number theory [30, 56, 33], spectral theory [44, 26, 48] and numerical analysis [25, 32, 24, 17]. Over the last decades it became a cornerstone of modeling in Engineering [65, 66, 51, 1, 27], and it can also be viewed today as a technique to regularize inverse potential problems [22, 31, 4]. Finding best rational approximants of prescribed degree to a specific function, say in the uniform norm on a given set, seems out of reach except in rare, particular cases. Indeed, such approximants depend in a rather convoluted manner, both on the approximated function and on the set where approximation should take place. Accordingly, the constructive side of the theory has focused on estimating optimal convergence rates as the degree grows large and devising approximation schemes coming close to meet them [67, 7, 23, 62], or else studying the behavior of natural, computationally appealing candidate approximants like Padé interpolants and their variants [3, 61, 45, 38, 42].

When a function is holomorphic in some neighborhood of a continuum $A$, the optimal speed of convergence for rational approximants on $A$ is at least geometric in the degree. Then, a coarse but manageable estimate of this speed proceeds via asymptotics of the $n$-th root of the error in approximation by rational functions of degree $n$. For functions analytically continuable off $A$ except over a polar set containing branchpoints (throughout *polar* means of logarithmic capacity zero), and provided that $A$ does not divide the extended complex plane, Gonchar and Rakhmanov constructed, using multipoint Padé interpolants and dwelling on work by the second author, a sequence of rational approximants whose $n$-th root error on $A$ is asymptotically the smallest possible. They further showed that these interpolants converge in capacity on the complement of a compact set $K$ minimizing the capacity of the condenser $(A, K)$ under the constraint that the function is analytic off $K$, and proved that the normalized counting measures of their poles converge to the condenser equilibrium distribution on $K$ [23]. It is remarkable that $K$ solves a geometric extremal problem from logarithmic potential theory, close in spirit to the Lavrentiev type [34], and that it depends merely on the set where approximation takes place, on the branchpoints of the approximated function and its monodromy, but nothing else. Such a structure emerges because only the $n$-th root of the error is considered, rather than the error itself. Since then, it has been an open issue whether any $n$-th root
optimal sequence of approximants – in particular a sequence of best approximants – has the same behavior. The present paper answers this question in the positive, at least when the branchpoints are finite in number and order. In particular, our results connect, apparently for the first time, the singularities of best uniform approximants to those of the approximated function. We also address the case of no branchpoints, when the approximated function is analytic except over a polar set, in which the speed of best rational approximation on $A$ is known to be faster than geometric with the degree. We prove that $n$-th root optimal approximants converge in capacity outside the singular set of the function, and that the “effective” poles converge in a sense to that set.

The gist of the paper becomes transparent upon observing that the behavior of rational approximants can be surmised when the function to be approximated extends analytically to a multiply-sheeted Riemann surface over the complex plane. As rational functions are single-valued, this topological discrepancy leads the approximants to mark out their domain of approximation by accumulating poles so as to form a cut, thereby preventing single-valued continuation in the limit. In the case of (diagonal) multipoint Padé interpolants, this cut has been characterized as being of smallest weighted capacity in a field that depends on the limiting distribution of the interpolation points, and poles asymptotically distribute according to the weighted equilibrium measure of that cut; moreover, the Padé interpolants converge in capacity on the extremal domain thus demarcated. This was established in [23], dwelling on the works [57, 58, 60, 61, 62] that deal with classical Padé interpolants and correspond to the zero field and unit weight; see also [46] for early developments along these lines, and [6] for applications to $L^2$-best rational approximation on the circle. Subsequently, by choosing interpolation points adequately and performing surgery to eliminate spurious poles, the authors of [23] construct, on any continuum $A$ not dividing the extended plane and contained in the analyticity domain of a function indefinitely continuable except over a closed polar set containing branchpoints, a sequence of rational approximants converging uniformly to that function on $A$ as their degree $n$ grows large and whose $n$-th root error has a *liminf* which is smallest possible, as well as a true limit. For this weakly optimal choice of interpolation points (meaning that the choice is optimal in the $n$-th root sense), the cut $K$ of minimum weighted capacity is also the cut minimizing the condenser capacity of $(A, K)$, as well as the cut of minimum Greenian capacity in the complement of $A$. The smallest value for the limit of the $n$-th root error is a simple, explicit function of this Greenian capacity, and the poles of the approximants thus constructed distribute asymptotically according to the Green equilibrium measure of that cut.

Now, assuming in addition that the branchpoints of the continuation off $A$ of the function to be approximated are finite in number and of algebraic type, we shall prove that *any* sequence of rational (or meromorphic) approximants of increasing degree $n$ whose $n$-th root error on $A$ converges to the smallest possible limit – a *fortiori* every sequence of best approximants – has the same asymptotic distribution of poles as the particular sequence constructed in [23]. More precisely, if a function analytic in a simply connected neighborhood of a continuum $A$ in the extended complex plane is indefinitely continuable off that neighborhood except over a closed polar set containing finitely many branchpoints, all of algebraic type, then *the normalized counting measures of the poles of any sequence of rational approximants of increasing degree $n$ with asymptotically optimal $n$-th root error on $A$ do converge weak-star, as $n$ grows large, to the Green equilibrium distribution of the compact set of minimum Greenian capacity outside of which the function is single-valued; moreover, convergence holds in capacity everywhere off that compact set*. Here, Green functions are understood with respect to the complement of the continuum $A$ where approximation takes place. Finally, if there are no branchpoints, that is, if the approximated function is single-valued and analytic on the extended complex plane except possibly over a closed polar set $E$, then there are rational approximants converging on $A$ faster than geometrically with the degree. We shall prove that such sequences of approximants (as well as their meromorphic analogs) converge in capacity on the extended plane deprived from $E$, and that retaining the singular part that comes close to $E$ generates new sequences of
optimal sequence of approximants – in particular a sequence of best approximants – has the same behavior. The present paper answers this question in the positive, at least when the branchpoints are finite in number and order. In particular, our results connect, apparently for the first time, the singularities of best uniform approximants to those of the approximated function. We also address the case of no branchpoints, when the approximated function is analytic except over a polar set, in which the speed of best rational approximation on A is known to be faster than geometric with the degree. We prove that n-th root optimal approximants converge in capacity outside the singular set of the function, and that the “effective” poles converge in a sense to that set.

The gist of the paper becomes transparent upon observing that the behavior of rational approximants can be surmised when the function to be approximated extends analytically to a multiply-sheeted Riemann surface over the complex plane. As rational functions are single-valued, this topological discrepancy leads the approximants to mark out their domain of approximation by accumulating poles so as to form a cut, thereby preventing single-valued continuation in the limit. In the case of (diagonal) multipoint Padé interpolants, this cut has been characterized as being of smallest weighted capacity in a field that depends on the limiting distribution of the interpolation points, and poles asymptotically distribute according to the weighted equilibrium measure of that cut; moreover, the Padé interpolants converge in capacity on the extremal domain thus demarcated. This was established in [23], dwelling on the works [57, 58, 60, 61, 62] that deal with classical Padé interpolants and correspond to the zero field and unit weight; see also [46] for early developments along these lines, and [6] for applications to $L^2$-best rational approximation on the circle. Subsequently, by choosing interpolation points adequately and performing surgery to eliminate spurious poles, the authors of [23] construct, on any continuum A not dividing the extended plane and contained in the analyticity domain of a function indefinitely continuable except over a closed polar set containing branchpoints, a sequence of rational approximants converging uniformly to that function on A as their degree $n$ grows large and whose $n$-th root error has a *liminf* which is smallest possible, as well as a true limit. For this weakly optimal choice of interpolation points (meaning that the choice is optimal in the $n$-th root sense), the cut $K$ of minimum weighted capacity is also the cut minimizing the condenser capacity of $(A, K)$, as well as the cut of minimum Greenian capacity in the complement of A. The smallest value for the limit of the $n$-th root error is a simple, explicit function of this Greenian capacity, and the poles of the approximants thus constructed distribute asymptotically according to the Green equilibrium measure of that cut.

Now, assuming in addition that the branchpoints of the continuation off A of the function to be approximated are finite in number and of algebraic type, we shall prove that *any* sequence of rational (or meromorphic) approximants of increasing degree $n$ whose $n$-th root error on A converges to the smallest possible limit – a *fortiori* every sequence of best approximants – has the same asymptotic distribution of poles as the particular sequence constructed in [23]. More precisely, if a function analytic in a simply connected neighborhood of a continuum A in the extended complex plane is indefinitely continuable off that neighborhood except over a closed polar set containing finitely many branchpoints, all of algebraic type, then *the normalized counting measures of the poles of any sequence of rational approximants of increasing degree n with asymptotically optimal n-th root error on A do converge weak-star, as n grows large, to the Green equilibrium distribution of the compact set of minimum Greenian capacity outside of which the function is single-valued; moreover, convergence holds in capacity everywhere off that compact set*. Here, Green functions are understood with respect to the complement of the continuum A where approximation takes place. Finally, if there are no branchpoints, that is, if the approximated function is single-valued and analytic on the extended complex plane except possibly over a closed polar set E, then there are rational approximants converging on A faster than geometrically with the degree. We shall prove that such sequences of approximants (as well as their meromorphic analogs) converge in capacity on the extended plane deprived from E, and that retaining the singular part that comes close to E generates new sequences of
approximants, still converging faster than geometrically with the degree, while satisfying in addition that any weak-star limit point of the normalized counting measures of their poles is supported on E.

Previously cited references, which deal with Padé or multipoint Padé interpolants, exploit the connection between denominators thereof and non-Hermitian orthogonal polynomials on a system of arcs encompassing the singular set of the interpolated function. Indeed, the core of the work in [60, 23] is to derive asymptotics of such polynomials on extremal systems of arcs like those constructed in [57, 58], so as to qualify the behavior of the poles of the interpolants when the degree grows large and deduce from it the desired convergence properties. Here, we proceed in the opposite direction: we assume that the optimal rate is met in the n-th root sense and deduce from it the behavior of the poles. For this we cannot make use of orthogonal polynomials, and in fact it is not even known if interpolation takes place at all in the case of best approximants. However, the construction from [57, 58] will still be basic to our purposes.

This work was initiated jointly by the three authors in 2009, but the untimely passing away of the second one on April 22nd, 2013 prevented him from seeing its completion. Still, some fundamental ideas are his.

## 2. PRELIMINARIES AND MAIN RESULTS

Given a function $f$ holomorphic in a neighborhood of a closed set $A \subset \bar{C}$, the error of approximation of $f$ on $A$ by rational functions of degree $n$ is

$$ (2.1) \qquad \rho_n(f, A) := \inf_{r \in \mathcal{R}_n(\bar{C} \setminus A)} \|f - r\|_A, $$

where $\| \cdot \|_A$ stands for the supremum norm on $A$ and, for $\Omega \subset \bar{C}$, we let $\mathcal{R}_n(\Omega)$ be the class of rational functions of type $(n, n)$ with all their poles in $\Omega$. That is, if $\mathcal{P}_n$ denotes the space of algebraic polynomials of degree at most $n$ and $M_n(\Omega)$ the monic polynomials of degree $n$ with all zeros in $\Omega$, then $\mathcal{R}_n(\Omega) := \mathcal{P}_n M_n^{-1}(\Omega)$. It was shown by Walsh [67, 2], using interpolation techniques, that

$$ (2.2) \qquad \limsup_{n \to \infty} \rho_n^{1/n}(f, A) \le \exp \{-1/\mathrm{cap}(A, K)\}, $$

where $K$ is any closed set disjoint from $A$ in the complement of which $f$ is holomorphic and $\mathrm{cap}(A, K)$ denotes the capacity of the condenser $(A, K)$. A definition of condenser capacity can be found in [55, Chapter VIII, Section 3]; for our purposes, it is enough to know that if $\bar{C} \setminus A$ is connected, then $\mathrm{cap}(A, K)$ coincides with the Greenian capacity $\mathrm{cap}_{\bar{C} \setminus A}(K)$ defined in Section A.4, see [55, Chapter VIII, Theorem 2.6 & Corollary 2.7] for this equivalence.

It is known that Walsh's inequality (2.2) cannot be improved [37]. On the other hand, it was conjectured by Gonchar [28] and proven by Parfènov when $A$ is a continuum with connected complement [47] (also later by Prokhorov for any compact set $A$ [52]) that

$$ (2.3) \qquad \liminf_{n \to \infty} \rho_n^{1/n}(f, A) \le \exp \{-2/\mathrm{cap}(A, K)\}. $$

Hence, $\rho_n(f, A)$ has no limit in general as $n \to \infty$, and when the limit exists it cannot exceed the right-hand side of (2.3). For certain classes of functions and certain loci of approximation $A$, it was nevertheless shown that $\rho_n(f, A)$ does have a limit, which is equal to the right-hand side of (2.3). More precisely, let $\mathcal{H}(A)$ denote the space of functions holomorphic on a (variable) neighborhood of $A$, and $\mathcal{S}(A) \subset \mathcal{H}(A)$ those functions continuable analytically into the complement of $A$ along any path that avoids some compact polar<fnref n="1" /> subset of $\bar{C} \setminus A$ (which may depend on the function); we require in addition that this continuation is not single-valued, namely that there are paths with the same endpoints leading to different analytic branches. Now, when $A$ is a continuum that does not

<footnote n="1">¹see Section A.5 for a definition and basic properties of polar sets, that may be defined as sets of zero logarithmic capacity.</footnote>
approximants, still converging faster than geometrically with the degree, while satisfying in addition that any weak-star limit point of the normalized counting measures of their poles is supported on E.

Previously cited references, which deal with Padé or multipoint Padé interpolants, exploit the connection between denominators thereof and non-Hermitian orthogonal polynomials on a system of arcs encompassing the singular set of the interpolated function. Indeed, the core of the work in [60, 23] is to derive asymptotics of such polynomials on extremal systems of arcs like those constructed in [57, 58], so as to qualify the behavior of the poles of the interpolants when the degree grows large and deduce from it the desired convergence properties. Here, we proceed in the opposite direction: we assume that the optimal rate is met in the n-th root sense and deduce from it the behavior of the poles. For this we cannot make use of orthogonal polynomials, and in fact it is not even known if interpolation takes place at all in the case of best approximants. However, the construction from [57, 58] will still be basic to our purposes.

This work was initiated jointly by the three authors in 2009, but the untimely passing away of the second one on April 22nd, 2013 prevented him from seeing its completion. Still, some fundamental ideas are his.

## 2. PRELIMINARIES AND MAIN RESULTS

Given a function $f$ holomorphic in a neighborhood of a closed set $A \subset \bar{C}$, the error of approximation of $f$ on $A$ by rational functions of degree $n$ is

$$ (2.1) \qquad \rho_n(f, A) := \inf_{r \in \mathcal{R}_n(\bar{C} \setminus A)} \|f - r\|_A, $$

where $\| \cdot \|_A$ stands for the supremum norm on $A$ and, for $\Omega \subset \bar{C}$, we let $\mathcal{R}_n(\Omega)$ be the class of rational functions of type $(n, n)$ with all their poles in $\Omega$. That is, if $\mathcal{P}_n$ denotes the space of algebraic polynomials of degree at most $n$ and $\mathcal{M}_n(\Omega)$ the monic polynomials of degree $n$ with all zeros in $\Omega$, then $\mathcal{R}_n(\Omega) := \mathcal{P}_n \mathcal{M}_n^{-1}(\Omega)$. It was shown by Walsh [67, 2], using interpolation techniques, that

$$ (2.2) \qquad \limsup_{n \to \infty} \rho_n^{1/n}(f, A) \le \exp \{-1/\mathrm{cap}(A, K)\}, $$

where $K$ is any closed set disjoint from $A$ in the complement of which $f$ is holomorphic and $\mathrm{cap}(A, K)$ denotes the capacity of the condenser $(A, K)$. A definition of condenser capacity can be found in [55, Chapter VIII, Section 3]; for our purposes, it is enough to know that if $\bar{C} \setminus A$ is connected, then $\mathrm{cap}(A, K)$ coincides with the Greenian capacity $\mathrm{cap}_{\bar{C} \setminus A}(K)$ defined in Section A.4, see [55, Chapter VIII, Theorem 2.6 & Corollary 2.7] for this equivalence.

It is known that Walsh's inequality (2.2) cannot be improved [37]. On the other hand, it was conjectured by Gonchar [28] and proven by Parfènov when $A$ is a continuum with connected complement [47] (also later by Prokhorov for any compact set $A$ [52]) that

$$ (2.3) \qquad \liminf_{n \to \infty} \rho_n^{1/n}(f, A) \le \exp \{-2/\mathrm{cap}(A, K)\}. $$

Hence, $\rho_n(f, A)$ has no limit in general as $n \to \infty$, and when the limit exists it cannot exceed the right-hand side of (2.3). For certain classes of functions and certain loci of approximation $A$, it was nevertheless shown that $\rho_n(f, A)$ does have a limit, which is equal to the right-hand side of (2.3). More precisely, let $\mathcal{H}(A)$ denote the space of functions holomorphic on a (variable) neighborhood of $A$, and $\mathcal{S}(A) \subset \mathcal{H}(A)$ those functions continuable analytically into the complement of $A$ along any path that avoids some compact polar<fnref n="1" /> subset of $\bar{C} \setminus A$ (which may depend on the function); we require in addition that this continuation is not single-valued, namely that there are paths with the same endpoints leading to different analytic branches. Now, when $A$ is a continuum that does not

<footnote n="1">¹see Section A.5 for a definition and basic properties of polar sets, that may be defined as sets of zero logarithmic capacity.</footnote>
separate the plane and $f \in S(A)$, it follows from work by the second author in [57, 58, 59] and it was explicitly stated by Gonchar and Rakhmanov in [23, Theorem 1'] that

$$ (2.4) \qquad \lim_{n \to \infty} \rho_n^{1/n}(f, A) = \inf_K \exp \{-2/\mathrm{cap}(A, K)\}, $$

where the infimum is taken over all compact sets $K$ such that $f$ admits a single-valued analytic continuation to $\bar{C} \setminus K$.

Hereafter, we let $T \subset \mathbb{C}$ be a Jordan curve with interior domain $D$, and we put $A := \bar{C} \setminus D$. In this setting $\infty \in A$, which is no loss of generality for a preliminary Möbius transform can ensure this; in contrast, our requirement that $D$ be a Jordan domain is a regularity assumption on the set $A$ where approximation takes place. Given $f \in \mathcal{H}(A)$, let $\mathcal{K}_f$ be the collection of all compact sets $K \subset D$ such that $f$, initially defined on $A$, admits a single-valued analytic continuation to $\bar{C} \setminus K$. It follows from [57, 58] that there exists $K_f \in \mathcal{K}_f$, unique up to addition and/or removal of a polar set, with

$$ (2.5) \qquad \mathrm{cap}_D(K_f) \le \mathrm{cap}_D(K), \quad K \in \mathcal{K}_f. $$

We can and will normalize $K_f$ to be the smallest possible, i.e., we make it the intersection of all $K \in \mathcal{K}_f$ for which $\mathrm{cap}_D(K)$ is minimal, see [58]. As $\mathrm{cap}(K, A) = \mathrm{cap}_D(K)$, in light of equation (2.4), our main goal is to investigate the asymptotic behavior of sequences $\{r_n\}$ of rational functions of type $(n, n)$ meeting this optimal $n$-th root rate:

$$ (2.6) \qquad \lim_{n \to \infty} \|f - r_n\|_A^{1/n} = \exp \{-2/\mathrm{cap}_D(K_f)\}. $$

We call any such sequence $\{r_n\}$ a sequence of $n$-th root optimal rational approximants to $f$ on $A$. In order to study $\{r_n\}$, we are led to consider more general sequences of meromorphic approximants of the form $r_n + h_n$, where $h_n$ is holomorphic in $D$ and continuous on $\bar{D}$, see Section 2.3 for the definitions. Even though best meromorphic approximants may look less natural than rational ones, they make contact with both the spectral theory of Hankel operators (through AAK theory) and Green potentials (because they generate errors with constant modulus on $T$), while remaining essentially equivalent to rational approximants as far as $n$-th root error rates are concerned [47]. This is why $n$-th root optimal meromorphic approximants (meeting (2.6) in place of $r_n$) are of principal importance in our study. Yet, the potential-theoretic tools on Riemann surfaces that we use only allow us to handle compact surfaces so far, and this induces some finiteness conditions on the functions from the class $S(A)$ that we can deal with. These are set forth in the next section.

## 2.1. Class of Approximated Functions.

(i) they can be continued into $D$ along any path originating on $T$ that stays in $\bar{D}$ while avoiding a closed polar subset of $D$ (which may depend on the function);

(ii) they are not single-valued, meaning there are continuations along at least two paths as in (i) with the same initial and terminal points that lead to distinct function elements, but still they are finite-valued in that the number of such function elements lying above a point of $D$ is uniformly bounded (the bound may depend on the function);

(iii) their number of branchpoints (points in any neighborhood of which some analytic continuation along a closed path in $D$ encircling that point while avoiding the exceptional polar set leads to a different function element) is finite.

In view of (i) and (ii), such functions lie in $S(A)$. Note that (iii) is not superfluous, for there are functions meeting (i) and (ii) with infinitely many branchpoints. For instance, an open Riemann surface $X$ made of two copies of $\bar{C} \setminus \{0\}$, glued along a sequence of disjoint cuts in $\mathbb{D}$ shrinking to the point 0, has projection $p: X \to \bar{C} \setminus \{0\}$ a two sheeted covering with infinitely many branchpoints of order 2. As $X$ carries a holomorphic function $f$ assuming more than one value on $p^{-1}(z)$ for $z$ not a critical value of $p$ [18, Theorem 26.7], we deduce on putting $D = \mathbb{D}$ and $A = \bar{C} \setminus \mathbb{D}$ that each branch of $f \circ p^{-1}$ is of the announced type.
separate the plane and $f \in S(A)$, it follows from work by the second author in [57, 58, 59] and it was explicitly stated by Gonchar and Rakhmanov in [23, Theorem 1'] that

$$ (2.4) \qquad \lim_{n \to \infty} \rho_n^{1/n}(f, A) = \inf_K \exp \{-2/cap(A, K)\}, $$

where the infimum is taken over all compact sets $K$ such that $f$ admits a single-valued analytic continuation to $\bar{C} \setminus K$.

Hereafter, we let $T \subset \mathbb{C}$ be a Jordan curve with interior domain $D$, and we put $A := \bar{C} \setminus D$. In this setting $\infty \in A$, which is no loss of generality for a preliminary Möbius transform can ensure this; in contrast, our requirement that $D$ be a Jordan domain is a regularity assumption on the set $A$ where approximation takes place. Given $f \in H(A)$, let $K_f$ be the collection of all compact sets $K \subset D$ such that $f$, initially defined on $A$, admits a single-valued analytic continuation to $\bar{C} \setminus K$. It follows from [57, 58] that there exists $K_f \in K$, unique up to addition and/or removal of a polar set, with

$$ (2.5) \qquad \mathrm{cap}_D(K_f) \le \mathrm{cap}_D(K), \quad K \in K_f. $$

We can and will normalize $K_f$ to be the smallest possible, i.e., we make it the intersection of all $K \in K_f$ for which $\mathrm{cap}_D(K)$ is minimal, see [58]. As $\mathrm{cap}(K, A) = \mathrm{cap}_D(K)$, in light of equation (2.4), our main goal is to investigate the asymptotic behavior of sequences $\{r_n\}$ of rational functions of type $(n, n)$ meeting this optimal $n$-th root rate:

$$ (2.6) \qquad \lim_{n \to \infty} \|f - r_n\|_A^{1/n} = \exp \{-2/cap_D(K_f)\}. $$

We call any such sequence $\{r_n\}$ a sequence of $n$-th root optimal rational approximants to $f$ on $A$. In order to study $\{r_n\}$, we are led to consider more general sequences of meromorphic approximants of the form $r_n + h_n$, where $h_n$ is holomorphic in $D$ and continuous on $\bar{D}$, see Section 2.3 for the definitions. Even though best meromorphic approximants may look less natural than rational ones, they make contact with both the spectral theory of Hankel operators (through AAK theory) and Green potentials (because they generate errors with constant modulus on $T$), while remaining essentially equivalent to rational approximants as far as $n$-th root error rates are concerned [47]. This is why $n$-th root optimal meromorphic approximants (meeting (2.6) in place of $r_n$) are of principal importance in our study. Yet, the potential-theoretic tools on Riemann surfaces that we use only allow us to handle compact surfaces so far, and this induces some finiteness conditions on the functions from the class $S(A)$ that we can deal with. These are set forth in the next section.

### 2.1. Class of Approximated Functions.

(i) they can be continued into $D$ along any path originating on $T$ that stays in $\bar{D}$ while avoiding a closed polar subset of $D$ (which may depend on the function);

(ii) they are not single-valued, meaning there are continuations along at least two paths as in (i) with the same initial and terminal points that lead to distinct function elements, but still they are finite-valued in that the number of such function elements lying above a point of $D$ is uniformly bounded (the bound may depend on the function);

(iii) their number of branchpoints (points in any neighborhood of which some analytic continuation along a closed path in $D$ encircling that point while avoiding the exceptional polar set leads to a different function element) is finite.

In view of (i) and (ii), such functions lie in $S(A)$. Note that (iii) is not superfluous, for there are functions meeting (i) and (ii) with infinitely many branchpoints. For instance, an open Riemann surface $X$ made of two copies of $\bar{C} \setminus \{0\}$, glued along a sequence of disjoint cuts in $D$ shrinking to the point $0$, has projection $p: X \to \bar{C} \setminus \{0\}$ a two sheeted covering with infinitely many branchpoints of order 2. As $X$ carries a holomorphic function $f$ assuming more than one value on $p^{-1}(z)$ for $z$ not a critical value of $p$ [18, Theorem 26.7], we deduce on putting $D = D$ and $A = \bar{C} \setminus D$ that each branch of $f \circ p^{-1}$ is of the announced type.
We formalize (i), (ii) and (iii) as follows. Let $\mathcal{R}_*$ be an auxiliary algebraic Riemann surface, whose set of ramification points $\mathbf{rp}(\mathcal{R}_*)$ lies on top of $D$. That is, there exists an irreducible polynomial in two variables $P(z, a)$, of degree at least 2 in $a$, such that $\mathcal{R}_* = \{(z, a) : P(z, a) = 0\}$ and all branchpoints of the algebraic function $a(z)$ lie in $D$. We denote by $p : \mathcal{R}_* \to \bar{\mathbb{C}}$ the natural projection $p((z, a)) = z$, and we let $\mathcal{R} \subset \mathcal{R}_*$ be the (open) Riemann surface defined as

$$ \mathcal{R} := \{z \in \mathcal{R}_* : p(z) \in D\}; $$

here and below, whenever it causes no confusion, we use letter $z$ to denote both points in $\mathbb{C}$ and on $\mathcal{R}_*$. Clearly, the ramification points of $\mathcal{R}_*$ and $\mathcal{R}$ are identical: $\mathbf{rp}(\mathcal{R}_*) = \mathbf{rp}(\mathcal{R})$. Let us denote by $\bar{\mathcal{R}}$ the closure of $\mathcal{R}$ in $\mathcal{R}_*$, and define a class of functions $\mathcal{F}(\mathcal{R})$ by

$$ (2.7) \qquad \mathcal{F}(\mathcal{R}) := \left\{ \begin{array}{ll} f & \text{f is holomorphic and single-valued on } \bar{\mathcal{R}}\setminus E_f, \\ E_f & \text{f is closed, } p(E_f) \text{ is polar and contained in } D, \\ f(z_1) \neq f(z_2) & \text{for some } z_1, z_2 \text{ with } p(z_1) = p(z_2). \end{array} \right. $$

In (2.7), we wrote $E_f$ for the singular set of $f$ on $\mathcal{R}$ but it would have been more appropriate to write $E_f(\mathcal{R})$, as the complete Riemann surface of $f$ could be significantly larger than $\mathcal{R}$ and its singular set bigger than $E_f$. Since all ramification points of $\mathcal{R}_*$ lie on top of $D$, the simple-connectedness of $A$ implies that the preimage $p^{-1}(T)$ consists of finitely many homeomorphic copies of $T$ under $p^{-1}$; we generically denote by $\mathcal{T}$ such a copy, so that $p : \mathcal{T} \to T$ is a homeomorphism. Then, denoting with a subscript $\lfloor T$ the restriction to a set $E$, the class of functions that we study is defined as

$$ (2.8) \qquad \mathcal{F}(A) := \left\{ f : f \text{ is holomorphic on } A \text{ and } \exists \mathcal{R}_*, \mathcal{T}, \hat{f} \in \mathcal{F}(\mathcal{R}) \text{ with } f_{\lfloor \mathcal{T}} = \hat{f} \circ (p_{\lfloor \mathcal{T}})^{-1} \right\}. $$

From (2.7) and (2.8), one sees that $\mathcal{F}(A) \subset \mathcal{S}(A) \subset \mathcal{H}(A)$ and members of $\mathcal{F}(A)$ meet (i), (ii), (iii). Conversely, when $f \in \mathcal{H}(A)$ satisfies (i), (ii) and (iii), one can check that $f \in \mathcal{F}(A)$. Indeed, if $B$ is the closed polar subset of $D$ defined by (i), we get from (ii) because $D\setminus B$ is connected, see Section A.5, that the number of sheets of the Riemann surface of $f$ above $D\setminus B$ is a finite constant, say $M$. Therefore, since the branchpoints are finitely many by (iii), the algebraic surface $\mathcal{R}_*$ can be constructed by a classical glueing process described in Section 3.2. The fine point, when applying to the present case this familiar procedure based on glueing pairwise in a certain order the banks of $M$ copies of a system of cuts joining the branchpoints, is that any two points of $D$ can be joined by a smooth simple arc entirely contained in $D\setminus B$, except for its endpoints if they lie in $B$. It is so because $D\setminus B$ is a connected open set and each point of $B$ is the center of a circle of arbitrary small radius contained in $D\setminus B$, as well as the endpoint of a segment contained in $D\setminus B$ (that may even be chosen to have quasi-any direction). These properties hold because $B$ is polar, and therefore thin at each point of $\mathbb{C}$, see Section A.6.

We also consider functions in $\mathcal{H}(A)$ meeting (i) but not (ii). These are analytic and single-valued in $\bar{\mathbb{C}}\setminus E$, where $E \subset D$ is closed and polar, i.e., there are no branchpoints. This case complements the previous one on putting $\mathcal{R}_* = \bar{\mathbb{C}}$ and omitting the last requirement in (2.7); we denote the corresponding class of functions by $\mathcal{E}(A)$. Since $\mathrm{cap}(A, E) = \mathrm{cap}_D(E) = 0$ when $E \subset D$ is polar, see Section A.4, we get from (2.3) and (2.2) that

$$ (2.9) \qquad \lim_{n \to \infty} \rho_n^{1/n}(f, A) = 0, \quad f \in \mathcal{E}(A). $$

That is to say, some sequence $\{r_n\}_{n \in \mathbb{N}}$, $r_n \in \mathcal{R}_n$, of rational functions, converges faster than geometrically towards $f$ on $A$ as the degree grows large, meaning that

$$ (2.10) \qquad \lim_{n \to \infty} \|f - r_n\|_A^{1/n} = 0. $$

We call any such sequence a sequence of n-th root optimal rational approximants to $f \in \mathcal{E}(A)$ on $A$.
We formalize (i), (ii) and (iii) as follows. Let $\mathcal{R}_*$ be an auxiliary algebraic Riemann surface, whose set of ramification points $\mathbf{rp}(\mathcal{R}_*)$ lies on top of $D$. That is, there exists an irreducible polynomial in two variables $P(z, a)$, of degree at least 2 in $a$, such that $\mathcal{R}_* = \{(z, a) : P(z, a) = 0\}$ and all branchpoints of the algebraic function $a(z)$ lie in $D$. We denote by $p : \mathcal{R}_* \to \bar{\mathbb{C}}$ the natural projection $p((z, a)) = z$, and we let $\mathcal{R} \subset \mathcal{R}_*$ be the (open) Riemann surface defined as

$$ \mathcal{R} := \{z \in \mathcal{R}_* : p(z) \in D\}; $$

here and below, whenever it causes no confusion, we use letter $z$ to denote both points in $\mathbb{C}$ and on $\mathcal{R}_*$. Clearly, the ramification points of $\mathcal{R}_*$ and $\mathcal{R}$ are identical: $\mathbf{rp}(\mathcal{R}_*) = \mathbf{rp}(\mathcal{R})$. Let us denote by $\bar{\mathcal{R}}$ the closure of $\mathcal{R}$ in $\mathcal{R}_*$, and define a class of functions $\mathcal{F}(\mathcal{R})$ by

$$ (2.7) \qquad \mathcal{F}(\mathcal{R}) := \left\{ \begin{array}{ll} f & \text{f is holomorphic and single-valued on } \bar{\mathcal{R}}\setminus E_f, \\ E_f & \text{f is closed, } p(E_f) \text{ is polar and contained in } D, \\ f(z_1) \neq f(z_2) & \text{for some } z_1, z_2 \text{ with } p(z_1) = p(z_2). \end{array} \right. $$

In (2.7), we wrote $E_f$ for the singular set of $f$ on $\mathcal{R}$ but it would have been more appropriate to write $E_f(\mathcal{R})$, as the complete Riemann surface of $f$ could be significantly larger than $\mathcal{R}$ and its singular set bigger than $E_f$. Since all ramification points of $\mathcal{R}_*$ lie on top of $D$, the simple-connectedness of $A$ implies that the preimage $p^{-1}(T)$ consists of finitely many homeomorphic copies of $T$ under $p^{-1}$; we generically denote by $\mathcal{T}$ such a copy, so that $p : \mathcal{T} \to T$ is a homeomorphism. Then, denoting with a subscript $\lfloor T$ the restriction to a set $E$, the class of functions that we study is defined as

$$ (2.8) \qquad \mathcal{F}(A) := \left\{ f : f \text{ is holomorphic on } A \text{ and } \exists \mathcal{R}_*, \mathcal{T}, \hat{f} \in \mathcal{F}(\mathcal{R}) \text{ with } f_{\lfloor \mathcal{T}} = \hat{f} \circ (p_{\lfloor \mathcal{T}})^{-1} \right\}. $$

From (2.7) and (2.8), one sees that $\mathcal{F}(A) \subset \mathcal{S}(A) \subset \mathcal{H}(A)$ and members of $\mathcal{F}(A)$ meet (i), (ii), (iii). Conversely, when $f \in \mathcal{H}(A)$ satisfies (i), (ii) and (iii), one can check that $f \in \mathcal{F}(A)$. Indeed, if $B$ is the closed polar subset of $D$ defined by (i), we get from (ii) because $D\setminus B$ is connected, see Section A.5, that the number of sheets of the Riemann surface of $f$ above $D\setminus B$ is a finite constant, say $M$. Therefore, since the branchpoints are finitely many by (iii), the algebraic surface $\mathcal{R}_*$ can be constructed by a classical glueing process described in Section 3.2. The fine point, when applying to the present case this familiar procedure based on glueing pairwise in a certain order the banks of $M$ copies of a system of cuts joining the branchpoints, is that any two points of $D$ can be joined by a smooth simple arc entirely contained in $D\setminus B$, except for its endpoints if they lie in $B$. It is so because $D\setminus B$ is a connected open set and each point of $B$ is the center of a circle of arbitrary small radius contained in $D\setminus B$, as well as the endpoint of a segment contained in $D\setminus B$ (that may even be chosen to have quasi-any direction). These properties hold because $B$ is polar, and therefore thin at each point of $\mathbb{C}$, see Section A.6.

We also consider functions in $\mathcal{H}(A)$ meeting (i) but not (ii). These are analytic and single-valued in $\bar{\mathbb{C}}\setminus E$, where $E \subset D$ is closed and polar, i.e., there are no branchpoints. This case complements the previous one on putting $\mathcal{R}_* = \bar{\mathbb{C}}$ and omitting the last requirement in (2.7); we denote the corresponding class of functions by $\mathcal{E}(A)$. Since $\mathrm{cap}(A, E) = \mathrm{cap}_D(E) = 0$ when $E \subset D$ is polar, see Section A.4, we get from (2.3) and (2.2) that

$$ (2.9) \qquad \lim_{n \to \infty} \rho_n^{1/n}(f, A) = 0, \quad f \in \mathcal{E}(A). $$

That is to say, some sequence $\{r_n\}_{n \in \mathbb{N}}$, $r_n \in \mathcal{R}_n$, of rational functions, converges faster than geometrically towards $f$ on $A$ as the degree grows large, meaning that

$$ (2.10) \qquad \lim_{n \to \infty} \|f - r_n\|_A^{1/n} = 0. $$

We call any such sequence a sequence of n-th root optimal rational approximants to $f \in \mathcal{E}(A)$ on $A$.
2.2. **Optimal Rational Approximants.** Notions of potential theory in $D$ and $\mathcal{R}$ play a fundamental role in what follows, and the reader might want to consult Appendix A for a comprehensive account thereof. Let us here recall the definition of Green potentials and Green equilibrium distributions.

The Green function $g_D(\cdot, w)$ of the domain $D$ with pole at $w \in D$ is the unique non-negative harmonic function in $D \setminus \{w\}$, with logarithmic singularity at $w$, whose largest harmonic minorant is zero. The Green potential of a positive Borel measure $v$ in $D$ is $g(v, D; z) := \int g_D(z, w)dv(w)$. Putting $|v|$ for the total mass of $v$, the Greenian capacity (in $D$) of a Borel set $B \subset D$ is defined by

$$ (2.11) \quad \mathrm{cap}_D(B) := \left( \inf_{v \ge 0, |v|=1, \mathrm{supp}\,v \subseteq B} I_D(v) \right)^{-1}, \quad I_D(v) := \iint g(z, w)dv(w)dv(z); $$

the infimum above is taken over all probability Borel measures $v$ supported on $B$. For any set $S \subset D$, the outer Greenian capacity of $S$ in $D$ is defined as $\mathrm{cap}_D(S) = \inf \mathrm{cap}_D(U)$, where the infimum is taken over all open sets $U \supset S$ in $D$ (using again the symbol $\mathrm{cap}_D$ causes no confusion for the outer Greenian capacity is known to coincide with the Greenian capacity on Borel sets). Polar subsets of $D$ are those whose outer Greenian capacity is 0. If $K$ is a non-polar compact subset of $D$, then there exists a unique Borel probability measure $\mu_{D,K}$ supported on $K$, called the Green equilibrium distribution on $K$ relative to $D$, such that $\mathrm{cap}_D(K) = 1/I_D(\mu_{D,K})$. It is characterized by the property that its Green potential is bounded on $D$ and equal to its maximum (which is then necessarily $1/\mathrm{cap}_D(K)$) quasi everywhere (that is, up to a polar set) on $K$. To recap:

$$ (2.12) \quad g(\mu_{D,K}, D; z) \begin{cases} \le 1/\mathrm{cap}_D(K), & z \in D, \\ = 1/\mathrm{cap}_D(K), & \text{for q.e. } z \in K, \\ < 1/\mathrm{cap}_D(K), & z \in D \setminus K, \end{cases} $$

where the last inequality follows from the (generalized) maximum principle for harmonic functions.

We are concerned with two types of asymptotics for sequences of rational approximants: the weak* behavior of the normalized counting measures of their poles, and the convergence in capacity of the functions themselves. More precisely, given a rational function $r$ of type $(n, n)$, we define

$$ (2.13) \quad \mu(r) := \frac{1}{n} \sum_{z:r(z)=\infty} \delta_z, $$

where each pole $z$ appears in the sum as many times as its order. Equivalently $-2\pi\mu(r)$ is the distributional Laplacian $\Delta(\log|q|^{1/n})$, with $q$ the denominator of an irreducible form of $r$. One says that a sequence $\{v_n\}$ of finite Borel measures on $D$ converges weak* to a measure $v$, denoted as

$$ v_n \xrightarrow{w^*} v \quad \text{as} \quad n \to \infty, $$

if $\int hdv_n \to \int hdv$ for every continuous compactly supported function $h$ on $D$. We further say that a sequence of functions $h_n$ converges in Greenian capacity to a function $h$ on a set $U \subset D$ if

$$ (2.14) \quad \lim_{n \to \infty} \mathrm{cap}_D \left( \{z \in F : |h_n(z) - h(z)| > \epsilon\} \right) = 0 $$

for each $\epsilon > 0$ and every compact $F \subset U$; we denote this claim by $h_n \xrightarrow{\mathrm{cap}} h$ in $U$. The convergence (2.14) is said to hold at a geometric rate if we can replace $\epsilon$ in (2.14) by $a^n$ for some positive $a = a(F) < 1$; the convergence rate is called faster than geometric if (2.14) holds with $\epsilon$ replaced by $a^n$ for any $a > 0$.

**Theorem 2.1.** Let $T \subset \mathbb{C}$ be a Jordan curve, $D$ its interior and $A$ the complement of $D \cup T$ on the Riemann sphere. Given $f \in \mathcal{F}(A)$, let $K_f \in \mathcal{K}_f$ be such that (2.5) holds and $\{r_n\}$ be a sequence of $n$-th root optimal rational approximants to $f$ on $A$, as defined in (2.6). Then,

$$ (2.15) \quad \mu(r_n) \xrightarrow{w^*} \mu_{D,K_f}, $$
2.2. **Optimal Rational Approximants.** Notions of potential theory in $D$ and $\mathcal{R}$ play a fundamental role in what follows, and the reader might want to consult Appendix A for a comprehensive account thereof. Let us here recall the definition of Green potentials and Green equilibrium distributions.

The Green function $g_D(\cdot, w)$ of the domain $D$ with pole at $w \in D$ is the unique non-negative harmonic function in $D \setminus \{w\}$, with logarithmic singularity at $w$, whose largest harmonic minorant is zero. The Green potential of a positive Borel measure $v$ in $D$ is $g(v, D; z) := \int g_D(z, w) dv(w)$. Putting $|v|$ for the total mass of $v$, the Greenian capacity (in $D$) of a Borel set $B \subset D$ is defined by

$$ (2.11) \quad \mathrm{cap}_D(B) := \left( \inf_{v \ge 0, |v|=1, \mathrm{supp}\,v \subseteq B} I_D(v) \right)^{-1}, \quad I_D(v) := \iint g(z, w) dv(w) dv(z); $$

the infimum above is taken over all probability Borel measures $v$ supported on $B$. For any set $S \subset D$, the outer Greenian capacity of $S$ in $D$ is defined as $\mathrm{cap}_D(S) = \inf \mathrm{cap}_D(U)$, where the infimum is taken over all open sets $U \supset S$ in $D$ (using again the symbol $\mathrm{cap}_D$ causes no confusion for the outer Greenian capacity is known to coincide with the Greenian capacity on Borel sets). Polar subsets of $D$ are those whose outer Greenian capacity is 0. If $K$ is a non-polar compact subset of $D$, then there exists a unique Borel probability measure $\mu_{D,K}$ supported on $K$, called the Green equilibrium distribution on $K$ relative to $D$, such that $\mathrm{cap}_D(K) = 1/I_D(\mu_{D,K})$. It is characterized by the property that its Green potential is bounded on $D$ and equal to its maximum (which is then necessarily $1/\mathrm{cap}_D(K)$) quasi everywhere (that is, up to a polar set) on $K$. To recap:

$$ (2.12) \quad g(\mu_{D,K}, D; z) \begin{cases} \le 1/\mathrm{cap}_D(K), & z \in D, \\ = 1/\mathrm{cap}_D(K), & \text{for q.e. } z \in K, \\ < 1/\mathrm{cap}_D(K), & z \in D \setminus K, \end{cases} $$

where the last inequality follows from the (generalized) maximum principle for harmonic functions.

We are concerned with two types of asymptotics for sequences of rational approximants: the weak* behavior of the normalized counting measures of their poles, and the convergence in capacity of the functions themselves. More precisely, given a rational function $r$ of type $(n, n)$, we define

$$ (2.13) \quad \mu(r) := \frac{1}{n} \sum_{z:r(z)=\infty} \delta_z, $$

where each pole $z$ appears in the sum as many times as its order. Equivalently $-2\pi\mu(r)$ is the distributional Laplacian $\Delta(\log|q|^{1/n})$, with $q$ the denominator of an irreducible form of $r$. One says that a sequence $\{v_n\}$ of finite Borel measures on $D$ converges weak* to a measure $v$, denoted as

$$ v_n \xrightarrow{w^*} v \quad \text{as} \quad n \to \infty, $$

if $\int hdv_n \to \int hdv$ for every continuous compactly supported function $h$ on $D$. We further say that a sequence of functions $h_n$ converges in Greenian capacity to a function $h$ on a set $U \subset D$ if

$$ (2.14) \quad \lim_{n \to \infty} \mathrm{cap}_D \left( \{z \in F : |h_n(z) - h(z)| > \epsilon\} \right) = 0 $$

for each $\epsilon > 0$ and every compact $F \subset U$; we denote this claim by $h_n \xrightarrow{\mathrm{cap}} h$ in $U$. The convergence (2.14) is said to hold at a geometric rate if we can replace $\epsilon$ in (2.14) by $a^n$ for some positive $a = a(F) < 1$; the convergence rate is called faster than geometric if (2.14) holds with $\epsilon$ replaced by $a^n$ for any $a > 0$.

**Theorem 2.1.** Let $T \subset \mathbb{C}$ be a Jordan curve, $D$ its interior and $A$ the complement of $D \cup T$ on the Riemann sphere. Given $f \in \mathcal{F}(A)$, let $K_f \in \mathcal{K}_f$ be such that (2.5) holds and $\{r_n\}$ be a sequence of $n$-th root optimal rational approximants to $f$ on $A$, as defined in (2.6). Then,

$$ (2.15) \quad \mu(r_n) \xrightarrow{w^*} \mu_{D,K_f}, $$
where $\mu(r_n)$ is the normalized counting measure of the poles of $r_n$ and $\mu_{D,K_f}$ is the Green equilibrium distribution on $K_f$ relative to $D$. Furthermore, it holds that

$$ (2.16) \qquad \frac{1}{2n} \log |f - r_n| \stackrel{\text{cap}}{\to} g(\mu_{D,K_f}, D; \cdot) - \frac{1}{\text{cap}_D(K_f)} \quad \text{in } D\setminus K_f \quad \text{as } n \to \infty, $$

where $g(\mu_{D,K_f}, D; \cdot)$ is the Green potential of $\mu_{D,K_f}$ in $D$.

**Remark 2.2.** Theorem 2.1 and equation (2.12) imply that $n$-th root optimal rational approximants converge to $f$ in capacity in $D\setminus K_f$, at a geometric rate equal to $\exp\{g(\mu_{D,K_f}, D; \cdot) - 1/\text{cap}_D(K_f)\}$ pointwise. In fact, as shown in Section 3.11, convergence in Greenian capacity in $D\setminus K_f$ and uniform convergence on $A$, along with the limiting behavior (2.15) for the poles, together imply the seemingly stronger convergence in logarithmic capacity on $\bar{D}\setminus K_f$, at a geometric rate which is pointwise no slower than the previous one. Here, convergence in logarithmic capacity is defined as in (2.14), except that the logarithmic capacity replaces the Greenian one, see Section A.4 for a definition of logarithmic capacity. This remark equally applies to the forthcoming Theorem 2.5, as well as to Theorems 2.3 and 2.6 in which $K_f$ gets replaced by a polar set $E$ and convergence takes place at faster than geometric rate. Moreover, if in Theorem 2.5 the error $f - M_n$ is asymptotically circular on $T$, namely if $\lim_n \inf_T |f - M_n|^{1/n} = \exp\{-2/\text{cap}_D(K_f)\}$ (in particular when $M_n$ are their own Nehari modifications, see Section 3.3), then one can draw the more precise conclusion that the exact pointwise rate of geometric convergence of $M_n$ to $f$ in logarithmic capacity on $\bar{D}\setminus K_f$ is the same as in Greenian capacity on $D\setminus K_f$, namely $\exp\{g(\mu_{D,K_f}, D; \cdot) - 1/\text{cap}_D(K_f)\}$.

<figure><img src="image_1.png" /><figcaption>FIGURE 1. Surface $\mathcal{R}$ with five ramification points $rp(\mathcal{R}) = \{a_1, a_2, a_3, a_4, a_5\}$ ($a_1$ is not labeled) and four sheets. Ramification point $a_1$ has order 3 and the remaining points have order 2. The sequence $①-②-③-④-①$ represents the monodromy around $a_1$ while encircling it clockwise, where the transitions happen across the dashed curves that stand for the cuts between different sheets of $\mathcal{R}$. The domain $U_f$ is depicted as a shaded region. In this example $E_0 = \emptyset$, $E_{11} = \{p(a_1), p(a_2), p(a_3)\}$ are the active branch points, and $E_{10} = \{b\}$, where $b = p(b_1) = p(b_2) = p(b_3)$. The set $K_f$ is a threefold equal to the natural projection of the solid (purple) curves. The solid curves also represent a different choice of the transition cuts between different sheets of $\mathcal{R}$, with the latter the domain $U_f$ will lie entirely on one sheet.</figcaption></figure>

In view of its importance, let us describe in greater detail the set $K_f$. As mentioned before, the problem of finding elements of minimal capacity in $K_f$ was extensively studied by the second author
where $\mu(r_n)$ is the normalized counting measure of the poles of $r_n$ and $\mu_{D,K_f}$ is the Green equilibrium distribution on $K_f$ relative to $D$. Furthermore, it holds that

$$ (2.16) \qquad \frac{1}{2n} \log |f - r_n| \stackrel{\text{cap}}{\rightarrow} g(\mu_{D,K_f}, D; \cdot) - \frac{1}{\text{cap}_D(K_f)} \quad \text{in } D\setminus K_f \quad \text{as } n \rightarrow \infty, $$

where $g(\mu_{D,K_f}, D; \cdot)$ is the Green potential of $\mu_{D,K_f}$ in $D$.

**Remark 2.2.** Theorem 2.1 and equation (2.12) imply that n-th root optimal rational approximants converge to $f$ in capacity in $D\setminus K_f$, at a geometric rate equal to $\exp\{g(\mu_{D,K_f}, D; \cdot) - 1/\text{cap}_D(K_f)\}$ pointwise. In fact, as shown in Section 3.11, convergence in Greenian capacity in $D\setminus K_f$ and uniform convergence on $A$, along with the limiting behavior (2.15) for the poles, together imply the seemingly stronger convergence in logarithmic capacity on $\bar{D}\setminus K_f$, at a geometric rate which is pointwise no slower than the previous one. Here, convergence in logarithmic capacity is defined as in (2.14), except that the logarithmic capacity replaces the Greenian one, see Section A.4 for a definition of logarithmic capacity. This remark equally applies to the forthcoming Theorem 2.5, as well as to Theorems 2.3 and 2.6 in which $K_f$ gets replaced by a polar set $E$ and convergence takes place at faster than geometric rate. Moreover, if in Theorem 2.5 the error $f - M_n$ is asymptotically circular on $T$, namely if $\lim_n \inf_T |f - M_n|^{1/n} = \exp\{-2/\text{cap}_D(K_f)\}$ (in particular when $M_n$ are their own Nehari modifications, see Section 3.3), then one can draw the more precise conclusion that the exact pointwise rate of geometric convergence of $M_n$ to $f$ in logarithmic capacity on $\bar{D}\setminus K_f$ is the same as in Greenian capacity on $D\setminus K_f$, namely $\exp\{g(\mu_{D,K_f}, D; \cdot) - 1/\text{cap}_D(K_f)\}$.

<figure><img src="image_3.png" /><figcaption>FIGURE 1. Surface $\mathcal{R}$ with five ramification points $rp(\mathcal{R}) = \{a_1, a_2, a_3, a_4, a_5\}$ ($a_1$ is not labeled) and four sheets. Ramification point $a_1$ has order 3 and the remaining points have order 2. The sequence ①-②-③-④-① represents the monodromy around $a_1$ while encircling it clockwise, where the transitions happen across the dashed curves that stand for the cuts between different sheets of $\mathcal{R}$. The domain $U_f$ is depicted as a shaded region. In this example $E_0 = \emptyset$, $E_{11} = \{p(a_1), p(a_2), p(a_3)\}$ are the active branch points, and $E_{10} = \{b\}$, where $b = p(b_1) = p(b_2) = p(b_3)$. The set $K_f$ is a threefold equal to the natural projection of the solid (purple) curves. The solid curves also represent a different choice of the transition cuts between different sheets of $\mathcal{R}$, with the latter the domain $U_f$ will lie entirely on one sheet.</figcaption></figure>

In view of its importance, let us describe in greater detail the set $K_f$. As mentioned before, the problem of finding elements of minimal capacity in $K_f$ was extensively studied by the second author
[57, 58, 59]. The existence of such sets for $f \in \mathcal{H}(A)$ was proven in [57], and one can choose $K_f$ so that $K_f \subseteq \tilde{K}$ for any $\tilde{K} \in \mathcal{K}_f$ with $\mathrm{cap}_D(\tilde{K}) = \inf_K \mathrm{cap}_D(K)$, which makes $K_f$ unique [58]. The topological structure of $K_f$ was investigated in [59], where it is shown that

$$ (2.17) \qquad K_f = E_0 \cup E_1 \cup \bigcup_i J_i $$

where the $J_i$ are open analytic arcs, $E_1$ comprises the endpoints of the arcs $J_i$, and $E_0$ is a subset of the singular set of $f$ in $D$ (the singular set consists of those points in $D$ to which some continuation of $f$ from $T$ has a singularity). As soon as $f \in \mathcal{S}(A)$, the set $E_0$ is polar by definition. To understand this decomposition better when $f \in \mathcal{F}(A)$, let $J_f := \cup_i \bar{J}_i$ so that $E_1 \subset J_f$. Then $f$ possesses a single-valued continuation into $D \setminus J_f$ with singular set $E_0$, consisting of polar and essential singularities (but no branching singularities). If $\mathcal{T}$, $\hat{f}$ are as in (2.8) and $\mathcal{U}_f$ is the connected component of $p^{-1}(D \setminus J_f)$ containing $\mathcal{T}$ in its boundary, then we can further decompose:

$$ E_0 = p(E_{\hat{f}} \cap \mathcal{U}_f), \quad E_1 = E_{10} \cup E_{11}, \quad E_{11} := p(\mathbf{rp}(\mathcal{R}) \cap \bar{\mathcal{U}}_f), \quad E_{10} := E_1 \setminus E_{11}. $$

That is, $E_{11}$ is the set of “active” branchpoints of $f$, i.e., branchpoints that can be reached by continuation of $f$ from $T$ into $D \setminus J_f$, as opposed to those points in $\mathbf{rp}(\mathcal{R})$ that cannot be so reached. Also, each $e \in E_{10}$ is an endpoint of at least three arcs $J_i$, and generically $f$ possesses analytic continuations to $e$ from any direction within $D \setminus J_f$ (unless by chance $e$ is a singularity of $f$ as well), see Figure 1. As $\mathbf{rp}(\mathcal{R})$ is finite, so is the collection $\{J_i\}$ in cases that we consider.

For $f \in \mathcal{F}(A)$, Theorem 2.1 asserts two things: (i) the weak* convergence of $\mu(r_n)$ to $\mu_{D,K_f}$ whenever $\{r_n\}$ is an $n$-th root optimal sequence of rational approximants to $f$ on $A$, and (ii) the convergence in capacity of $r_n$ to $f$ at a geometric rate on $D \setminus K_f$. If now $f \in \mathcal{E}(A)$ and $\{r_n\}$ is a sequence of $n$-th root optimal rational approximants to $f$ on $A$, i.e., a sequence meeting (2.10) and thus converging faster than geometrically to $f$ on $A$, then we shall see that $r_n$ converges in capacity to $f$ at faster than geometric rate in $D$. However, one can no longer expect a specific behavior of $\mu(r_n)$ in this case, for if $R_n$ is a sequence in $\mathcal{R}_n(D)$ that converges to zero faster than geometrically on $T$, then $r_{n/2} + R_{n/2}$ is again a sequence of $n$-th root optimal rational approximants to $f$ on $A$, whereas the weak* limit points of $\mu(R_n)$ can be arbitrary amongst positive measures of mass at most 1 on $D$. Thus, faster than geometric convergence does not qualify rational approximants enough to imply much on the behavior of their poles. Still, those poles of $r_n$ that stay away from the singular set of $f$ cannot account for the rate of convergence. This is made precise in the following result, which complements Theorem 2.1 in the case of no branchpoints.

**Theorem 2.3.** Let $T$, $A$ and $D$ be as in Theorem 2.1. Given $f \in \mathcal{E}(A)$, let $\{r_n\}$ be a sequence of rational functions of type $(n, n)$ meeting (2.10). Then, it holds that

$$ r_n \xrightarrow{\mathrm{cap}} f \quad \text{in} \quad D \setminus E $$

at faster than geometric rate, where $E \subset D$ a closed polar set outside of which $f$ is analytic and single-valued. Moreover, for any neighborhood $V$ of $E$ there is a sequence $\{R_{k_n}\}$, $R_{k_n} \in \mathcal{R}_{k_n}(V)$, $k_n \le n$, such that the poles of $R_{k_n}$ are among the poles of $r_n$ lying in $V$ and

$$ (2.18) \qquad \lim_{n \to \infty} \|f - R_{k_n}\|_A^{1/n} = 0. $$

Using neighborhoods $V$ shrinking to $E$ and a diagonal argument, Theorem 2.3 yields a corollary of independent interest.

**Corollary 2.4.** If $f \in \mathcal{E}(A)$ and $E \subset D$ a closed polar set outside of which $f$ is analytic and single-valued, then there is a sequence of rational functions $\{r_n\}$, $r_n \in \mathcal{R}_n(D)$, converging faster than geometrically to $f$ on $A$, and such that every weak* cluster point of the sequence $\{\mu(r_n)\}$ of normalized counting measures of the poles is supported on $E$.
[57, 58, 59]. The existence of such sets for $f \in \mathcal{H}(A)$ was proven in [57], and one can choose $K_f$ so that $K_f \subseteq \tilde{K}$ for any $\tilde{K} \in \mathcal{K}_f$ with $\mathrm{cap}_D(\tilde{K}) = \inf_K \mathrm{cap}_D(K)$, which makes $K_f$ unique [58]. The topological structure of $K_f$ was investigated in [59], where it is shown that

$$ (2.17) \qquad K_f = E_0 \cup E_1 \cup \bigcup_i J_i $$

where the $J_i$ are open analytic arcs, $E_1$ comprises the endpoints of the arcs $J_i$, and $E_0$ is a subset of the singular set of $f$ in $D$ (the singular set consists of those points in $D$ to which some continuation of $f$ from $T$ has a singularity). As soon as $f \in \mathcal{S}(A)$, the set $E_0$ is polar by definition. To understand this decomposition better when $f \in \mathcal{F}(A)$, let $J_f := \cup_i \bar{J}_i$ so that $E_1 \subset J_f$. Then $f$ possesses a single-valued continuation into $D \setminus J_f$ with singular set $E_0$, consisting of polar and essential singularities (but no branching singularities). If $\mathcal{T}$, $\hat{f}$ are as in (2.8) and $\mathcal{U}_f$ is the connected component of $p^{-1}(D \setminus J_f)$ containing $\mathcal{T}$ in its boundary, then we can further decompose:

$$ E_0 = p(E_{\hat{f}} \cap \mathcal{U}_f), \quad E_1 = E_{10} \cup E_{11}, \quad E_{11} := p(\mathbf{rp}(\mathcal{R}) \cap \bar{\mathcal{U}}_f), \quad E_{10} := E_1 \setminus E_{11}. $$

That is, $E_{11}$ is the set of “active” branchpoints of $f$, i.e., branchpoints that can be reached by continuation of $f$ from $T$ into $D \setminus J_f$, as opposed to those points in $\mathbf{rp}(\mathcal{R})$ that cannot be so reached. Also, each $e \in E_{10}$ is an endpoint of at least three arcs $J_i$, and generically $f$ possesses analytic continuations to $e$ from any direction within $D \setminus J_f$ (unless by chance $e$ is a singularity of $f$ as well), see Figure 1. As $\mathbf{rp}(\mathcal{R})$ is finite, so is the collection $\{J_i\}$ in cases that we consider.

For $f \in \mathcal{F}(A)$, Theorem 2.1 asserts two things: (i) the weak* convergence of $\mu(r_n)$ to $\mu_{D,K_f}$ whenever $\{r_n\}$ is an $n$-th root optimal sequence of rational approximants to $f$ on $A$, and (ii) the convergence in capacity of $r_n$ to $f$ at a geometric rate on $D \setminus K_f$. If now $f \in \mathcal{E}(A)$ and $\{r_n\}$ is a sequence of $n$-th root optimal rational approximants to $f$ on $A$, i.e., a sequence meeting (2.10) and thus converging faster than geometrically to $f$ on $A$, then we shall see that $r_n$ converges in capacity to $f$ at faster than geometric rate in $D$. However, one can no longer expect a specific behavior of $\mu(r_n)$ in this case, for if $R_n$ is a sequence in $\mathcal{R}_n(D)$ that converges to zero faster than geometrically on $T$, then $r_{n/2} + R_{n/2}$ is again a sequence of $n$-th root optimal rational approximants to $f$ on $A$, whereas the weak* limit points of $\mu(R_n)$ can be arbitrary amongst positive measures of mass at most 1 on $D$. Thus, faster than geometric convergence does not qualify rational approximants enough to imply much on the behavior of their poles. Still, those poles of $r_n$ that stay away from the singular set of $f$ cannot account for the rate of convergence. This is made precise in the following result, which complements Theorem 2.1 in the case of no branchpoints.

**Theorem 2.3.** Let $T$, $A$ and $D$ be as in Theorem 2.1. Given $f \in \mathcal{E}(A)$, let $\{r_n\}$ be a sequence of rational functions of type $(n, n)$ meeting (2.10). Then, it holds that

$$ r_n \xrightarrow{\mathrm{cap}} f \quad \text{in} \quad D \setminus E $$

at faster than geometric rate, where $E \subset D$ a closed polar set outside of which $f$ is analytic and single-valued. Moreover, for any neighborhood $V$ of $E$ there is a sequence $\{R_{k_n}\}$, $R_{k_n} \in \mathcal{R}_{k_n}(V)$, $k_n \le n$, such that the poles of $R_{k_n}$ are among the poles of $r_n$ lying in $V$ and

$$ (2.18) \qquad \lim_{n \to \infty} \|f - R_{k_n}\|_A^{1/n} = 0. $$

Using neighborhoods $V$ shrinking to $E$ and a diagonal argument, Theorem 2.3 yields a corollary of independent interest.

**Corollary 2.4.** If $f \in \mathcal{E}(A)$ and $E \subset D$ a closed polar set outside of which $f$ is analytic and single-valued, then there is a sequence of rational functions $\{r_n\}$, $r_n \in \mathcal{R}_n(D)$, converging faster than geometrically to $f$ on $A$, and such that every weak* cluster point of the sequence $\{\mu(r_n)\}$ of normalized counting measures of the poles is supported on $E$.
2.3. **Optimal Meromorphic Approximants.** Let $H^\infty(D)$ denote the space of bounded analytic functions on $D$ and $\mathcal{A}(D)$ the subspace of those extending continuously to $\bar{D}$. When $T$ is rectifiable each $h \in H^\infty(D)$ has a non-tangential limit almost everywhere on $T$ with respect to arclength, that we still call $h$, and putting $\| \cdot \|_T$ for the essential supremum norm on $T$ (with respect to arclength) it holds that $\|h\|_T = \|h\|_D$ [12, Theorems 10.3 & 10.5]. When $T$ is a non-rectifiable Jordan curve, however, limiting values of $H^\infty(D)$-functions on $T$ generally exist at sectorially accessible points only, and such points may reduce to a set of zero linear measure [41]. This will force us into a somewhat careful discussion of meromorphic approximants. Remember the set $M_n(D)$ of monic polynomials of degree $n$ whose zeros belong to $D$, and put

$$ \begin{cases} H_n^\infty(D) & := \{h/q : h \in H^\infty(D), q \in M_n(D)\}, \\ \mathcal{A}_n(D) & := \{h/q : h \in \mathcal{A}(D), q \in M_n(D)\}. \end{cases} $$

That is, $H_n^\infty(D)$ is the set of meromorphic function with at most $n$ poles in $D$ that are bounded near $T$, and $\mathcal{A}_n(D)$ is the subset of those extending continuously to $T$. We shall say that $\{M_n\}$ is a sequence of $n$-th root optimal meromorphic approximants to $f \in S(A)$ on $T$ if $M_n \in \mathcal{A}_n(D)$ and

$$ (2.19) \qquad \limsup_{n \to \infty} \|f - M_n\|_T^{1/n} \le \exp\{-2/\mathrm{cap}_D(K_f)\}. $$

Any sequence of $n$-th root optimal rational approximants is a particular sequence of $n$-th root optimal meromorphic ones, by (2.6). When $T$ is rectifiable, Corollary 2.7 to come will entail that in (2.19) the condition $M_n \in \mathcal{A}_n(D)$ can be traded for the seemingly weaker requirement $M_n \in H_n^\infty(D)$. However, it is important to require that $M_n \in \mathcal{A}_n(D)$ when $T$ is not rectifiable, for otherwise the left-hand side of (2.19) may no longer make sense for the essential supremum norm (with respect to arclength) and the considerations below would not apply. In his proof of (2.3), Parfënov [47] has shown that the limit inferior of $\rho_n^{1/n}(f, A)$ is the same as the one of the $n$-th root of the error in best meromorphic approximation, see (2.22) for a definition of best (not just $n$-th root optimal) meromorphic approximants. Hence, in light of (2.6), replacing rational approximants with meromorphic ones does not improve the rate of decay of the $n$-th root error, and $n$-th root optimal meromorphic approximants to functions in $S(A)$ could be defined with the limit superior and the inequality sign replaced by a full limit and the equality sign in (2.19).

**Theorem 2.5.** With the notation of Theorem 2.1, let $\{M_n\}$ be a sequence of $n$-th root optimal meromorphic approximants to $f \in \mathcal{F}(A)$ on $T$, see (2.19). Then, the conclusions of Theorem 2.1 hold with $r_n$ replaced by $M_n$.

Let us reiterate that $n$-th root optimal rational approximants are just an instance of $n$-th root optimal meromorphic approximants, and therefore Theorem 2.1 is a special case of Theorem 2.5.

When $f \in \mathcal{E}(A)$, we define meromorphic approximants in a manner similar to (2.19) but this time with an eye on (2.9). Namely, we say that $\{M_n\}$ is a sequence of $n$-th root optimal meromorphic approximants to $f \in \mathcal{E}(A)$ on $T$ if $M_n \in \mathcal{A}_n(D)$ and

$$ (2.20) \qquad \lim_{n \to \infty} \|f - M_n\|_T^{1/n} = 0. $$

The following complements Theorem 2.5 in the case of no branchpoints, and subsumes Theorem 2.3.

**Theorem 2.6.** With the notation of Theorem 2.3, let $\{M_n\}$ be a sequence of $n$-th root optimal meromorphic approximants on $T$ to $f \in \mathcal{E}(A)$, see (2.20). Then, the conclusions of Theorem 2.3 hold with $r_n$ replaced by $M_n$.

Besides best rational approximants, a noteworthy instance of $n$-th root optimal meromorphic approximants are the AAK (short for Adamyan-Arov-Krein) approximants. These are best meromorphic approximants with at most $n$ poles. Specifically, consider the following (Nehari-Takagi)
2.3. **Optimal Meromorphic Approximants.** Let $H^\infty(D)$ denote the space of bounded analytic functions on $D$ and $\mathcal{A}(D)$ the subspace of those extending continuously to $\bar{D}$. When $T$ is rectifiable each $h \in H^\infty(D)$ has a non-tangential limit almost everywhere on $T$ with respect to arclength, that we still call $h$, and putting $\| \cdot \|_T$ for the essential supremum norm on $T$ (with respect to arclength) it holds that $\|h\|_T = \|h\|_D$ [12, Theorems 10.3 & 10.5]. When $T$ is a non-rectifiable Jordan curve, however, limiting values of $H^\infty(D)$-functions on $T$ generally exist at sectorially accessible points only, and such points may reduce to a set of zero linear measure [41]. This will force us into a somewhat careful discussion of meromorphic approximants. Remember the set $M_n(D)$ of monic polynomials of degree $n$ whose zeros belong to $D$, and put

$$ \left\{ \begin{array}{l} H_n^\infty(D) := \{h/q : h \in H^\infty(D), q \in M_n(D)\}, \\ \mathcal{A}_n(D) := \{h/q : h \in \mathcal{A}(D), q \in M_n(D)\}. \end{array} \right. $$

That is, $H_n^\infty(D)$ is the set of meromorphic function with at most $n$ poles in $D$ that are bounded near $T$, and $\mathcal{A}_n(D)$ is the subset of those extending continuously to $T$. We shall say that $\{M_n\}$ is a sequence of $n$-th root optimal meromorphic approximants to $f \in S(A)$ on $T$ if $M_n \in \mathcal{A}_n(D)$ and

$$ (2.19) \qquad \limsup_{n \to \infty} \|f - M_n\|_T^{1/n} \le \exp \{-2/cap_D(K_f)\}. $$

Any sequence of $n$-th root optimal rational approximants is a particular sequence of $n$-th root optimal meromorphic ones, by (2.6). When $T$ is rectifiable, Corollary 2.7 to come will entail that in (2.19) the condition $M_n \in \mathcal{A}_n(D)$ can be traded for the seemingly weaker requirement $M_n \in H_n^\infty(D)$. However, it is important to require that $M_n \in \mathcal{A}_n(D)$ when $T$ is not rectifiable, for otherwise the left-hand side of (2.19) may no longer make sense for the essential supremum norm (with respect to arclength) and the considerations below would not apply. In his proof of (2.3), Parfënov [47] has shown that the limit inferior of $\rho_n^{1/n}(f, A)$ is the same as the one of the $n$-th root of the error in best meromorphic approximation, see (2.22) for a definition of best (not just $n$-th root optimal) meromorphic approximants. Hence, in light of (2.6), replacing rational approximants with meromorphic ones does not improve the rate of decay of the $n$-th root error, and $n$-th root optimal meromorphic approximants to functions in $S(A)$ could be defined with the limit superior and the inequality sign replaced by a full limit and the equality sign in (2.19).

**Theorem 2.5.** With the notation of Theorem 2.1, let $\{M_n\}$ be a sequence of $n$-th root optimal meromorphic approximants to $f \in \mathcal{F}(A)$ on $T$, see (2.19). Then, the conclusions of Theorem 2.1 hold with $r_n$ replaced by $M_n$.

Let us reiterate that $n$-th root optimal rational approximants are just an instance of $n$-th root optimal meromorphic approximants, and therefore Theorem 2.1 is a special case of Theorem 2.5.

When $f \in \mathcal{E}(A)$, we define meromorphic approximants in a manner similar to (2.19) but this time with an eye on (2.9). Namely, we say that $\{M_n\}$ is a sequence of $n$-th root optimal meromorphic approximants to $f \in \mathcal{E}(A)$ on $T$ if $M_n \in \mathcal{A}_n(D)$ and

$$ (2.20) \qquad \lim_{n \to \infty} \|f - M_n\|_T^{1/n} = 0. $$

The following complements Theorem 2.5 in the case of no branchpoints, and subsumes Theorem 2.3.

**Theorem 2.6.** With the notation of Theorem 2.3, let $\{M_n\}$ be a sequence of $n$-th root optimal meromorphic approximants on $T$ to $f \in \mathcal{E}(A)$, see (2.20). Then, the conclusions of Theorem 2.3 hold with $r_n$ replaced by $M_n$.

Besides best rational approximants, a noteworthy instance of $n$-th root optimal meromorphic approximants are the AAK (short for Adamyan-Arov-Krein) approximants. These are best meromorphic approximants with at most $n$ poles. Specifically, consider the following (Nehari-Takagi)
problem: given $f \in L^\infty(\mathbb{T})$, find $M_n^\infty \in H_n^\infty(\mathbb{D})$ such that

$$ (2.21) \qquad \|f - M_n^\infty\|_{\mathbb{T}} = \inf_{M \in H_n^\infty(\mathbb{D})} \|f - M\|_{\mathbb{T}}. $$

If $n=0$, then (2.21) reduces to the question of best analytic approximation of bounded functions on the unit circle by elements of $H^\infty(\mathbb{D})$, which is the so-called Nehari problem named after [43] (that deals with an equivalent issue). It is known that $M_n^\infty$ always exists and that it is unique when $f$ lies in $C(\mathbb{T}) + H^\infty(\mathbb{D})$, see [48, Chapter 4]. Moreover, if $f$ is Dini-continuous on $\mathbb{T}$, then $M_n^\infty$ is continuous on $\mathbb{T}$. Indeed, if we write $M_n^\infty = r_n + g$ where $r_n \in \mathcal{R}_n(\mathbb{D})$ and $g \in H^\infty(\mathbb{D})$, we see that $g$ must be the best Nehari approximant to $f - r_n$, which is Dini-continuous on $\mathbb{T}$, and the claim follows from [9]. When $T$ is a rectifiable Jordan curve, one can readily replace $\mathbb{D}$ by $D$ in (2.21) and carry over to $T$ all the properties of best meromorphic approximants on $\mathbb{T}$ by conformal mapping. When $T$ is non-rectifiable, the very existence of approximants depends on the analyticity of $f$ on $T$, and follows from the proof of the next corollary to Theorem 2.5.

**Corollary 2.7.** Let $A, B$ and $D$ be as in Theorem 2.1. Given $f \in \mathcal{F}(A)$ or $\mathcal{E}(A)$, to each integer $n$ there exists a unique $M_n^\infty \in \mathcal{A}_n(D)$ such that

$$ (2.22) \qquad \|f - M_n^\infty\|_{\mathbb{T}} = \inf_{M \in \mathcal{A}_n(D)} \|f - M\|_{\mathbb{T}}, $$

and if $T$ is rectifiable then $\mathcal{A}_n(D)$ can be replaced by $H_n^\infty(D)$ in (2.22) without changing $M_n^\infty$. Of necessity, the conclusions of Theorem 2.5 and 2.6 hold with $M_n$ replaced by $M_n^\infty$.

Much interest in best meromorphic approximants stems from their striking connection to operator theory. Denote by $L^2(\mathbb{T})$ the space of square integrable functions on $\mathbb{T}$, and let $H^2 \subset L^2(\mathbb{T})$ be the Hardy space of functions whose Fourier coefficients with negative index do vanish. It is known that $H^2$ can be identified with (non-tangential limits on $\mathbb{T}$ of) analytic functions in $\mathbb{D}$ whose $L^2$-means on circles centered at the origin are uniformly bounded, see [12, Theorem 3.4]. Set $H^2_\perp := L^2(\mathbb{T}) \ominus H^2$ to be the orthogonal complement of $H^2$, which is the Hardy space of $L^2$-functions whose Fourier coefficients with nonnegative index are equal to zero. The latter can be identified with functions analytic in $\bar{\mathbb{C}}\setminus\bar{\mathbb{D}}$ that vanish at infinity, and whose $L^2$-means with respect to normalized arclength on circles centered at the origin are uniformly bounded. Let $\mathbb{P}_\perp : L^2(\mathbb{T}) \to H^2_\perp$ be the orthogonal projection. Given $f \in L^\infty(\mathbb{T})$, one defines the *Hankel operator with symbol f* to be

$$ (2.23) \qquad \Gamma_f : H^2 \to H^2_\perp, \qquad \Gamma_f(g) := \mathbb{P}_\perp(gf). $$

For $n$ a non-negative integer, let $s_n(\Gamma_f)$ be the $(n+1)$-th singular number of the operator $\Gamma_f$, that is $s_n(\Gamma_f) := \inf_{\text{rank } R \le n} \|\Gamma_f - R\|$, where the infimum is taken over all operators $R : H^2 \to H^2_\perp$ of rank at most $n$ and $\| \cdot \|$ stands for the operator norm. Then, one has that

$$ (2.24) \qquad \|f - M_n^\infty\|_{\mathbb{T}} = s_n(\Gamma_f). $$

If, in addition, $f \in C(\mathbb{T}) + H^\infty(\mathbb{D})$, then $\Gamma_f$ is compact so that $s_n^2(\Gamma_f)$ is the $(n+1)$-st eigenvalue of $\Gamma_f^*\Gamma_f$ when these are arranged in non-increasing order, and (2.24) becomes a pointwise equality:

$$ (2.25) \qquad |(f - M_n^\infty)(z)| = s_n(\Gamma_f) \quad \text{a.e. on } \mathbb{T}. $$

Moreover, if $v_n$ is an $(n+1)$-st singular vector of $\Gamma_f$, i.e., an eigenvector of $\Gamma_f^*\Gamma_f$ with eigenvalue $s_n^2(\Gamma_f)$, then $M_n^\infty$ is explicitly given in terms of $f$ and $v_n$ by the formula

$$ (2.26) \qquad f - M_n^\infty = \frac{\Gamma_f(v_n)}{v_n}. $$

Though not obvious at first glance, the right-hand side of (2.26) is independent of which $n$-th singular vector $v_n$ is chosen and it has constant modulus on $\mathbb{T}$, in accordance with (2.25). The next corollary, of independent interest, follows from (2.25), (2.19), (2.9), and Theorems 2.5–2.6.
problem: given $f \in L^\infty(\mathbb{T})$, find $M_n^\infty \in H_n^\infty(\mathbb{D})$ such that

$$ (2.21) \qquad \|f - M_n^\infty\|_{\mathbb{T}} = \inf_{M \in H_n^\infty(\mathbb{D})} \|f - M\|_{\mathbb{T}}. $$

If $n = 0$, then (2.21) reduces to the question of best analytic approximation of bounded functions on the unit circle by elements of $H^\infty(\mathbb{D})$, which is the so-called Nehari problem named after [43] (that deals with an equivalent issue). It is known that $M_n^\infty$ always exists and that it is unique when $f$ lies in $C(\mathbb{T}) + H^\infty(\mathbb{D})$, see [48, Chapter 4]. Moreover, if $f$ is Dini-continuous on $\mathbb{T}$, then $M_n^\infty$ is continuous on $\mathbb{T}$. Indeed, if we write $M_n^\infty = r_n + g$ where $r_n \in \mathcal{R}_n(\mathbb{D})$ and $g \in H^\infty(\mathbb{D})$, we see that $g$ must be the best Nehari approximant to $f - r_n$, which is Dini-continuous on $\mathbb{T}$, and the claim follows from [9]. When $T$ is a rectifiable Jordan curve, one can readily replace $\mathbb{D}$ by $D$ in (2.21) and carry over to $T$ all the properties of best meromorphic approximants on $\mathbb{T}$ by conformal mapping. When $T$ is non-rectifiable, the very existence of approximants depends on the analyticity of $f$ on $T$, and follows from the proof of the next corollary to Theorem 2.5.

**Corollary 2.7.** Let $A, B$ and $D$ be as in Theorem 2.1. Given $f \in \mathcal{F}(A)$ or $\mathcal{E}(A)$, to each integer $n$ there exists a unique $M_n^\infty \in \mathcal{A}_n(D)$ such that

$$ (2.22) \qquad \|f - M_n^\infty\|_{\mathbb{T}} = \inf_{M \in \mathcal{A}_n(D)} \|f - M\|_{\mathbb{T}}, $$

and if $T$ is rectifiable then $\mathcal{A}_n(D)$ can be replaced by $H_n^\infty(D)$ in (2.22) without changing $M_n^\infty$. Of necessity, the conclusions of Theorem 2.5 and 2.6 hold with $M_n$ replaced by $M_n^\infty$.

Much interest in best meromorphic approximants stems from their striking connection to operator theory. Denote by $L^2(\mathbb{T})$ the space of square integrable functions on $\mathbb{T}$, and let $H^2 \subset L^2(\mathbb{T})$ be the Hardy space of functions whose Fourier coefficients with negative index do vanish. It is known that $H^2$ can be identified with (non-tangential limits on $\mathbb{T}$ of) analytic functions in $\mathbb{D}$ whose $L^2$-means on circles centered at the origin are uniformly bounded, see [12, Theorem 3.4]. Set $H^2_- := L^2(\mathbb{T}) \ominus H^2$ to be the orthogonal complement of $H^2$, which is the Hardy space of $L^2$-functions whose Fourier coefficients with nonnegative index are equal to zero. The latter can be identified with functions analytic in $\bar{\mathbb{C}}\setminus\bar{\mathbb{D}}$ that vanish at infinity, and whose $L^2$-means with respect to normalized arclength on circles centered at the origin are uniformly bounded. Let $\mathbb{P}_- : L^2(\mathbb{T}) \to H^2_-$ be the orthogonal projection. Given $f \in L^\infty(\mathbb{T})$, one defines the *Hankel operator with symbol f* to be

$$ (2.23) \qquad \Gamma_f : H^2 \to H^2_-, \qquad \Gamma_f(g) := \mathbb{P}_-(gf). $$

For $n$ a non-negative integer, let $s_n(\Gamma_f)$ be the $(n+1)$-th singular number of the operator $\Gamma_f$, that is $s_n(\Gamma_f) := \inf_{\text{rank } R \le n} \|\Gamma_f - R\|$, where the infimum is taken over all operators $R : H^2 \to H^2_-$ of rank at most $n$ and $\| \cdot \|$ stands for the operator norm. Then, one has that

$$ (2.24) \qquad \|f - M_n^\infty\|_{\mathbb{T}} = s_n(\Gamma_f). $$

If, in addition, $f \in C(\mathbb{T}) + H^\infty(\mathbb{D})$, then $\Gamma_f$ is compact so that $s_n^2(\Gamma_f)$ is the $(n+1)$-st eigenvalue of $\Gamma_f^*\Gamma_f$ when these are arranged in non-increasing order, and (2.24) becomes a pointwise equality:

$$ (2.25) \qquad |(f - M_n^\infty)(z)| = s_n(\Gamma_f) \quad \text{a.e. on } \mathbb{T}. $$

Moreover, if $v_n$ is an $(n+1)$-st singular vector of $\Gamma_f$, i.e., an eigenvector of $\Gamma_f^*\Gamma_f$ with eigenvalue $s_n^2(\Gamma_f)$, then $M_n^\infty$ is explicitly given in terms of $f$ and $v_n$ by the formula

$$ (2.26) \qquad f - M_n^\infty = \frac{\Gamma_f(v_n)}{v_n}. $$

Though not obvious at first glance, the right-hand side of (2.26) is independent of which $n$-th singular vector $v_n$ is chosen and it has constant modulus on $\mathbb{T}$, in accordance with (2.25). The next corollary, of independent interest, follows from (2.25), (2.19), (2.9), and Theorems 2.5–2.6.
**Corollary 2.8.** Let $\Gamma_f$ be the Hankel operator with symbol $f \in \mathcal{F}(\bar{\mathbb{C}}\setminus\mathbb{D})$. Then, it holds that

$$ \lim_{n \to \infty} s_n^{1/n} (\Gamma_f) = \exp \{-2/\mathrm{cap}_{\mathbb{D}}(K_f)\}. $$

Moreover, if $f \in \mathcal{E}(\bar{\mathbb{C}}\setminus\mathbb{D})$, then the above limit is equal to 0.

### 3. PROOF OF THEOREM 2.5

**3.1. Existence of Best Meromorphic Approximants.** Below, we establish the existence and uniqueness part of Corollary 2.7, along with the assertion that $H_n^\infty(D)$ may replace $\mathcal{A}_n(D)$ when $T$ is rectifiable. The rest of the corollary will follow from Theorem 2.5 upon conclusion of its proof.

Let $\phi: \mathbb{D} \to D$ be a conformal map. Since $D$ is a Jordan domain, $\phi$ extends to a homeomorphism from $\bar{\mathbb{D}}$ to $\bar{D}$ by Carathéodory theorem [50, Section 2]. Let $L^2(\mathbb{T})$, $H^2$, and $H_2^-$ be as defined after Corollary 2.7 and $P_+: L^2(\mathbb{T}) \to H^2$, $P_-: L^2(\mathbb{T}) \to H_2^-$ be the orthogonal projections. If we pick a continuous function $f$ on $T$, then $f \circ \phi$ is continuous on $T$ and, in particular, it lies in $L^2(\mathbb{T})$. Set $F := P_-(f \circ \phi)$ and $G := P_+(f \circ \phi)$, so that $F \in H_2^-$ and $G \in H^2$. Since $P_+ + P_-$ is the identity operator, it holds that

$$ (3.1) \qquad F(z) = (f \circ \phi)(z) - G(z), \quad \text{a.e. } z \in \mathbb{T}. $$

Now, if $f \in \mathcal{H}(A)$, then $f \circ \phi(z)$ is holomorphic in $r < |z| < 1$ and continuous in $r \le |z| \le 1$, for $r$ close enough to 1. Hence, the right-hand side of (3.1) is holomorphic in $r < |z| < 1$ with uniformly bounded $L^2$-means on circles centered at the origin, while the left-hand side lies in $H_2^-$ and both sides have the same non-tangential limit on $T$. By an easy variant of Morera's theorem [21, Chapter II, Exercise 12], the function equal to $F(z)$ for $|z| > 1$ and to $(f \circ \phi)(z) - G(z)$ for $r < |z| < 1$ is holomorphic across $T$, in particular $F$ extends analytically across $T$ and $G$ extends continuously to $\bar{\mathbb{D}}$. Then, as described after (2.21), the best meromorphic approximant $M_n^\infty \in H_n^\infty(D)$ to $f \circ \phi$ exists and is unique, moreover it is readily checked that $M_n^\infty$ is equal to the sum of $G$ (a member of $\mathcal{A}(D)$) and of the best approximant to $F$ from $H_n^\infty(D)$, which lies in $\mathcal{A}_n(D)$ because $F$ is analytic across $T$ and so is a fortiori Dini-continuous on $T$; hence, we get that $M_n^\infty \in \mathcal{A}_n(D)$. If now $M \in \mathcal{A}_n(D)$, then $M \circ \phi \in \mathcal{A}_n(D)$ and

$$ \|f - M\|_T = \|f \circ \phi - M \circ \phi\|_T \ge \|f \circ \phi - M_n^\infty\|_T $$

by definition of $M_n^\infty$. As $M_n^\infty \circ \phi^{-1} \in \mathcal{A}_n(D)$, it is the unique best meromorphic approximant to $f$ we are looking for. The previous argument also shows that, when $f \in \mathcal{H}(A)$, the best meromorphic approximant to $f \circ \phi$ necessarily belongs to $\mathcal{A}_n(D)$. Because composition with $\phi$ is an isometric isomorphism $L^\infty(T) \to L^\infty(\mathbb{T})$ (understood with respect to arclength measure) when $T$ is rectifiable, one can equivalently use $H_n^\infty(D)$ instead of $\mathcal{A}_n(D)$ in definition (2.22).

**3.2. Reduction to the Unit Disk.** Let $K$ be a compact subset of $\mathbb{D}$. Since Green potentials are non-negative superharmonic functions whose largest harmonic minorant is zero, while a characteristic property of Green equilibrium potentials is to be constant quasi everywhere on the support of their defining measure while being no greater than this constant everywhere in the domain, it holds that

$$ g(\mu_{\mathbb{D},K}, \mathbb{D}; z) = g(\mu_{\mathbb{D},\phi(K)}, \mathbb{D}; \phi(z)) \quad \text{and} \quad \mathrm{cap}_{\mathbb{D}}(K) = \mathrm{cap}_{\mathbb{D}}(\phi(K)) $$

where $\phi: \mathbb{D} \to D$ is a conformal map, see (2.12) as well as Sections A.3 and A.4. One has in this case that $\mu_{\mathbb{D},\phi(K)} = \phi_*(\mu_{\mathbb{D},K})$, the pushforward of $\mu_{\mathbb{D},K}$ under $\phi$. Hence, we get that

$$ \frac{1}{n} \sum_{i=1}^{n} \delta_{z_i} \stackrel{*}{\to} \mu_{\mathbb{D},K} \quad \text{if and only if} \quad \frac{1}{n} \sum_{i=1}^{n} \delta_{\phi(z_i)} \stackrel{*}{\to} \mu_{\mathbb{D},\phi(K)}, $$

where the weak* convergence is understood for $n \to \infty$. Furthermore, by conformal invariance, a sequence of functions $h_n$ converges in Greenian capacity to a function $h$ in $D$ if and only if the functions $h_n \circ \phi$ converge in Greenian capacity to the function $h \circ \phi$ in $\mathbb{D}$.
**Corollary 2.8.** Let $\Gamma_f$ be the Hankel operator with symbol $f \in \mathcal{F}(\bar{\mathbb{C}}\setminus\mathbb{D})$. Then, it holds that

$$ \lim_{n \to \infty} s_n^{1/n} (\Gamma_f) = \exp \{-2/\mathrm{cap}_{\mathbb{D}}(K_f)\}. $$

Moreover, if $f \in \mathcal{E}(\bar{\mathbb{C}}\setminus\mathbb{D})$, then the above limit is equal to 0.

### 3. PROOF OF THEOREM 2.5

**3.1. Existence of Best Meromorphic Approximants.** Below, we establish the existence and uniqueness part of Corollary 2.7, along with the assertion that $H_n^\infty(D)$ may replace $\mathcal{A}_n(D)$ when $T$ is rectifiable. The rest of the corollary will follow from Theorem 2.5 upon conclusion of its proof.

Let $\phi: \mathbb{D} \to D$ be a conformal map. Since $D$ is a Jordan domain, $\phi$ extends to a homeomorphism from $\bar{\mathbb{D}}$ to $\bar{D}$ by Carathéodory theorem [50, Section 2]. Let $L^2(\mathbb{T})$, $H^2$, and $H_2^-$ be as defined after Corollary 2.7 and $P_+ : L^2(\mathbb{T}) \to H^2$, $P_- : L^2(\mathbb{T}) \to H_2^-$ be the orthogonal projections. If we pick a continuous function $f$ on $T$, then $f \circ \phi$ is continuous on $T$ and, in particular, it lies in $L^2(\mathbb{T})$. Set $F := P_-(f \circ \phi)$ and $G := P_+(f \circ \phi)$, so that $F \in H_2^-$ and $G \in H^2$. Since $P_+ + P_-$ is the identity operator, it holds that

$$ (3.1) \qquad F(z) = (f \circ \phi)(z) - G(z), \quad \text{a.e. } z \in \mathbb{T}. $$

Now, if $f \in \mathcal{H}(A)$, then $f \circ \phi(z)$ is holomorphic in $r < |z| < 1$ and continuous in $r \le |z| \le 1$, for $r$ close enough to 1. Hence, the right-hand side of (3.1) is holomorphic in $r < |z| < 1$ with uniformly bounded $L^2$-means on circles centered at the origin, while the left-hand side lies in $H_2^-$ and both sides have the same non-tangential limit on $T$. By an easy variant of Morera's theorem [21, Chapter II, Exercise 12], the function equal to $F(z)$ for $|z| > 1$ and to $(f \circ \phi)(z) - G(z)$ for $r < |z| < 1$ is holomorphic across $T$, in particular $F$ extends analytically across $T$ and $G$ extends continuously to $\bar{\mathbb{D}}$. Then, as described after (2.21), the best meromorphic approximant $M_n^\infty \in H_n^\infty(D)$ to $f \circ \phi$ exists and is unique, moreover it is readily checked that $M_n^\infty$ is equal to the sum of $G$ (a member of $\mathcal{A}(D)$) and of the best approximant to $F$ from $H_n^\infty(D)$, which lies in $\mathcal{A}_n(D)$ because $F$ is analytic across $T$ and so is a fortiori Dini-continuous on $T$; hence, we get that $M_n^\infty \in \mathcal{A}_n(D)$. If now $M \in \mathcal{A}_n(D)$, then $M \circ \phi \in \mathcal{A}_n(D)$ and

$$ \|f - M\|_T = \|f \circ \phi - M \circ \phi\|_T \ge \|f \circ \phi - M_n^\infty\|_T $$

by definition of $M_n^\infty$. As $M_n^\infty \circ \phi^{-1} \in \mathcal{A}_n(D)$, it is the unique best meromorphic approximant to $f$ we are looking for. The previous argument also shows that, when $f \in \mathcal{H}(A)$, the best meromorphic approximant to $f \circ \phi$ necessarily belongs to $\mathcal{A}_n(D)$. Because composition with $\phi$ is an isometric isomorphism $L^\infty(T) \to L^\infty(\mathbb{T})$ (understood with respect to arclength measure) when $T$ is rectifiable, one can equivalently use $H_n^\infty(D)$ instead of $\mathcal{A}_n(D)$ in definition (2.22).

**3.2. Reduction to the Unit Disk.** Let $K$ be a compact subset of $\mathbb{D}$. Since Green potentials are non-negative superharmonic functions whose largest harmonic minorant is zero, while a characteristic property of Green equilibrium potentials is to be constant quasi everywhere on the support of their defining measure while being no greater than this constant everywhere in the domain, it holds that

$$ g(\mu_{\mathbb{D},K}, \mathbb{D}; z) = g(\mu_{\mathbb{D},\phi(K)}, \mathbb{D}; \phi(z)) \quad \text{and} \quad \mathrm{cap}_{\mathbb{D}}(K) = \mathrm{cap}_{\mathbb{D}}(\phi(K)) $$

where $\phi: \mathbb{D} \to D$ is a conformal map, see (2.12) as well as Sections A.3 and A.4. One has in this case that $\mu_{\mathbb{D},\phi(K)} = \phi_*(\mu_{\mathbb{D},K})$, the pushforward of $\mu_{\mathbb{D},K}$ under $\phi$. Hence, we get that

$$ \frac{1}{n} \sum_{i=1}^{n} \delta_{z_i} \stackrel{*}{\to} \mu_{\mathbb{D},K} \quad \text{if and only if} \quad \frac{1}{n} \sum_{i=1}^{n} \delta_{\phi(z_i)} \stackrel{*}{\to} \mu_{\mathbb{D},\phi(K)}, $$

where the weak* convergence is understood for $n \to \infty$. Furthermore, by conformal invariance, a sequence of functions $h_n$ converges in Greenian capacity to a function $h$ in $D$ if and only if the functions $h_n \circ \phi$ converge in Greenian capacity to the function $h \circ \phi$ in $\mathbb{D}$.
Let now $\mathcal{R}_*$ and $\mathcal{R}$ be as in Section 2.1. Denote by $M$ the number of sheets of $\mathcal{R}$, so that a generic point in $D$ has $M$ preimages under the natural projection $p: \mathcal{R} \to D$. Let $J \subset D$ be a smooth oriented Jordan arc joining two points of $p(\mathbf{rp}(\mathcal{R}))$ while passing through all others exactly once. Constructing $J$ is tantamount to enumerating the points of $p(\mathbf{rp}(\mathcal{R}))$. Write $J = \cup_l J_l$ where each $J_l$ connects exactly two points in $p(\mathbf{rp}(\mathcal{R}))$. The surface $\mathcal{R}$ can be realized as $M$ copies $U_1, \dots, U_M$ of $D \setminus J$, suitably glued to each other along the banks of the cuts $J_l$ in each copy $U_i$ (the glueing rule can be encoded, for example, as a collection of 4-tuples $(l, i, j, k)$ telling one that the left and right banks of the cut $J_l$ in $U_i$ need to be glued to the right bank of the cut $J_l$ in $U_j$ and the left bank of the cut $J_l$ in $U_k$ respectively, see [40] for a discussion of Hurwitz's theorem). Using the same gluing rule, we can construct another $M$-sheeted surface $S$ out of the domains $\phi^{-1}(U_1), \dots, \phi^{-1}(U_M)$. The map $\phi$ can then be lifted to a conformal map $\Phi: S \to \mathcal{R}$ for which $\phi(\pi(z)) = p(\Phi(z))$, $z \in S$, where $\pi: S \to D$ is the natural projection.

Let $f \in \mathcal{F}(A)$ and $\hat{f} \in \mathcal{F}(\mathcal{R})$ be as in (2.8) and (2.7). Clearly $\hat{f} \circ \Phi \in \mathcal{F}(S)$, and the surface $S_*$ can be constructed by gluing $M$ copies of $\overline{\mathbb{C}} \setminus D$ to $S$ along the $M$ homeomorphic copies of $T$ that comprise the boundary of $S$. Arguing as we did after (3.1), we find that $F := \mathbb{P}_-(f \circ \phi)$ lies in $\mathcal{F}(\overline{\mathbb{C}} \setminus D)$ with the corresponding $\hat{F} \in \mathcal{F}(S)$ given by $\hat{f} \circ \Phi - G \circ \pi$, where $G := \mathbb{P}_+(f \circ \phi)$. Note that the conformal equivalence of Greenian capacities implies that $K_F = \phi^{-1}(K_f)$. Hence, if $\{M_n\}$ is an $n$-th root optimal sequence of meromorphic approximants of $f$ in $D$ as defined in (2.19), then $\{\tilde{M}_n := M_n \circ \phi - G\}$ is an $n$-th root optimal sequence of meromorphic approximants to $F$ on $T$ and

$$ (3.2) \qquad (f - M_n)(\phi(z)) = (F - \tilde{M}_n)(z), \quad z \in \overline{D} \setminus K_F. $$

Therefore, it is sufficient to study the asymptotic behavior of $F - \tilde{M}_n$ as well as the limit distribution of poles of $\tilde{M}_n$. That is, it is enough to prove Theorem 2.5 on the unit disk.

### 3.3. Nehari Modifications.
For $f \in \mathcal{H}(\overline{\mathbb{C}} \setminus D)$ and $M_n \in \mathcal{A}_n(D)$, let $h_{f-M_n}$ be the best holomorphic (Nehari) approximant of $f-M_n$ in $H^\infty(D)$. That is, $\|f-M_n-h_{f-M_n}\|_T = \inf_{h \in H^\infty(D)} \|f-M_n-h\|_T$, and $h_{f-M_n} \in H^\infty(D)$. Let us set

$$ (3.3) \qquad N(M_n)(z) := (M_n + h_{f-M_n})(z), $$

and call $N(M_n)$ the Nehari modification of $M_n$. The discussion after (2.21) shows that $N(M_n)$ lies in $\mathcal{A}_n(D)$. Indeed, we may write $M_n = g_n + r_n$ with $r_n \in \mathcal{R}_n(D)$ and $g_n \in \mathcal{A}(D)$. Then, one can readily check that $h_{f-M_n} = g_n + h_{f-r_n}$. As $f-r_n$ is analytic across $T$ and in particular Dini-continuous there, it follows that $h_{f-r_n}$ belongs to $\mathcal{A}(D)$ and so does $h_{f-M_n}$.

Since $\|f - N(M_n)\|_T \le \|f - M_n\|_T$ and $N(M_n)$ lies in $\mathcal{A}_n(D)$, the sequence $\{N(M_n)\}$ is one of $n$-th root optimal meromorphic approximants to $f$, whenever $\{M_n\}$ is such a sequence. It is beneficial for us to consider Nehari modifications because they enjoy the additional property that the error they generate has constant modulus on $T$, i.e., it follows from (2.24) that

$$ (3.4) \qquad |(f - N(M_n))(z)| = \|\Gamma_{f-M_n}\| \quad \text{for a.e. } z \in T. $$

We claim that it is enough to prove Theorem 2.5 for Nehari modifications only, as we now show.

Assume that Theorem 2.5 holds for $\{N(M_n)\}$. As the poles of $M_n$ and $N(M_n)$ are the same, this automatically yields the statement about weak* convergence of the counting measures of the poles. Moreover, given $\epsilon > 0$ and $K \subset D \setminus K_f$ a compact set, let us put

$$ E(K, \epsilon, N(M_n)) := \left\{ z \in K : \left| \frac{1}{2n} \log |(f - N(M_n))(z)| - g(\mu_{D,K_f}, D; z) + \frac{1}{\text{cap}_D(K_f)} \right| > \epsilon \right\}. $$

Define $E(K, \epsilon, M_n)$ analogously. According to our assumption it holds that

$$ (3.5) \qquad \lim_{n \to \infty} \text{cap}_D(E(K, \epsilon, N(M_n))) = 0, $$
Let now $\mathcal{R}_*$ and $\mathcal{R}$ be as in Section 2.1. Denote by $M$ the number of sheets of $\mathcal{R}$, so that a generic point in $D$ has $M$ preimages under the natural projection $p: \mathcal{R} \to D$. Let $J \subset D$ be a smooth oriented Jordan arc joining two points of $p(\mathbf{rp}(\mathcal{R}))$ while passing through all others exactly once. Constructing $J$ is tantamount to enumerating the points of $p(\mathbf{rp}(\mathcal{R}))$. Write $J = \cup_l J_l$ where each $J_l$ connects exactly two points in $p(\mathbf{rp}(\mathcal{R}))$. The surface $\mathcal{R}$ can be realized as $M$ copies $U_1, \dots, U_M$ of $D \setminus J$, suitably glued to each other along the banks of the cuts $J_l$ in each copy $U_i$ (the glueing rule can be encoded, for example, as a collection of 4-tuples $(l, i, j, k)$ telling one that the left and right banks of the cut $J_l$ in $U_i$ need to be glued to the right bank of the cut $J_l$ in $U_j$ and the left bank of the cut $J_l$ in $U_k$ respectively, see [40] for a discussion of Hurwitz's theorem). Using the same gluing rule, we can construct another $M$-sheeted surface $S$ out of the domains $\phi^{-1}(U_1), \dots, \phi^{-1}(U_M)$. The map $\phi$ can then be lifted to a conformal map $\Phi: S \to \mathcal{R}$ for which $\phi(\pi(z)) = p(\Phi(z))$, $z \in S$, where $\pi: S \to D$ is the natural projection.

Let $f \in \mathcal{F}(A)$ and $\hat{f} \in \mathcal{F}(\mathcal{R})$ be as in (2.8) and (2.7). Clearly $\hat{f} \circ \Phi \in \mathcal{F}(S)$, and the surface $S_*$ can be constructed by gluing $M$ copies of $\overline{\mathbb{C}} \setminus D$ to $S$ along the $M$ homeomorphic copies of $\mathbb{T}$ that comprise the boundary of $S$. Arguing as we did after (3.1), we find that $F := \mathbb{P}_-(f \circ \phi)$ lies in $\mathcal{F}(\overline{\mathbb{C}} \setminus D)$ with the corresponding $\hat{F} \in \mathcal{F}(S)$ given by $\hat{f} \circ \Phi - G \circ \pi$, where $G := \mathbb{P}_+(f \circ \phi)$. Note that the conformal equivalence of Greenian capacities implies that $K_F = \phi^{-1}(K_f)$. Hence, if $\{M_n\}$ is an $n$-th root optimal sequence of meromorphic approximants of $f$ in $D$ as defined in (2.19), then $\{\tilde{M}_n := M_n \circ \phi - G\}$ is an $n$-th root optimal sequence of meromorphic approximants to $F$ on $\mathbb{T}$ and

$$ (3.2) \qquad (f - M_n)(\phi(z)) = (F - \tilde{M}_n)(z), \quad z \in \overline{D} \setminus K_F. $$

Therefore, it is sufficient to study the asymptotic behavior of $F - \tilde{M}_n$ as well as the limit distribution of poles of $\tilde{M}_n$. That is, it is enough to prove Theorem 2.5 on the unit disk.

### 3.3. Nehari Modifications.
For $f \in \mathcal{H}(\overline{\mathbb{C}} \setminus D)$ and $M_n \in \mathcal{A}_n(D)$, let $h_{f-M_n}$ be the best holomorphic (Nehari) approximant of $f-M_n$ in $H^\infty(D)$. That is, $\|f-M_n-h_{f-M_n}\|_{\mathbb{T}} = \inf_{h \in H^\infty(D)} \|f-M_n-h\|_{\mathbb{T}}$, and $h_{f-M_n} \in H^\infty(D)$. Let us set

$$ (3.3) \qquad N(M_n)(z) := (M_n + h_{f-M_n})(z), $$

and call $N(M_n)$ the Nehari modification of $M_n$. The discussion after (2.21) shows that $N(M_n)$ lies in $\mathcal{A}_n(D)$. Indeed, we may write $M_n = g_n + r_n$ with $r_n \in \mathcal{R}_n(D)$ and $g_n \in \mathcal{A}(D)$. Then, one can readily check that $h_{f-M_n} = g_n + h_{f-r_n}$. As $f-r_n$ is analytic across $\mathbb{T}$ and in particular Dini-continuous there, it follows that $h_{f-r_n}$ belongs to $\mathcal{A}(D)$ and so does $h_{f-M_n}$.

Since $\|f - N(M_n)\|_{\mathbb{T}} \le \|f - M_n\|_{\mathbb{T}}$ and $N(M_n)$ lies in $\mathcal{A}_n(D)$, the sequence $\{N(M_n)\}$ is one of $n$-th root optimal meromorphic approximants to $f$, whenever $\{M_n\}$ is such a sequence. It is beneficial for us to consider Nehari modifications because they enjoy the additional property that the error they generate has constant modulus on $\mathbb{T}$, i.e., it follows from (2.24) that

$$ (3.4) \qquad |(f - N(M_n))(z)| = \|\Gamma_{f-M_n}\| \quad \text{for a.e. } z \in \mathbb{T}. $$

We claim that it is enough to prove Theorem 2.5 for Nehari modifications only, as we now show.

Assume that Theorem 2.5 holds for $\{N(M_n)\}$. As the poles of $M_n$ and $N(M_n)$ are the same, this automatically yields the statement about weak* convergence of the counting measures of the poles. Moreover, given $\epsilon > 0$ and $K \subset D \setminus K_f$ a compact set, let us put

$$ E(K, \epsilon, N(M_n)) := \left\{ z \in K : \left| \frac{1}{2n} \log |(f - N(M_n))(z)| - g(\mu_{D,K_f}, D; z) + \frac{1}{\text{cap}_D(K_f)} \right| > \epsilon \right\}. $$

Define $E(K, \epsilon, M_n)$ analogously. According to our assumption it holds that

$$ (3.5) \qquad \lim_{n \to \infty} \text{cap}_D(E(K, \epsilon, N(M_n))) = 0, $$
and we need to show that (3.5) holds with $N(M_n)$ replaced by $M_n$. Given $\varepsilon \in (0, 1)$, define

$$F_{n, \varepsilon} := \left\{ z \in K : \frac{1}{2n} \log |(f - N(M_n))(z)| > g(\mu_{\mathbb{D},K_f}, \mathbb{D}; \cdot) - \frac{1}{\operatorname{cap}_{\mathbb{D}}(K_f)} - \varepsilon m_K \right\},$$

where $m_K := \min_K g(\mu_{\mathbb{D},K_f}, \mathbb{D}; \cdot) > 0$. Since $N(M_n) - M_n$ is analytic in $\mathbb{D}$, we get from the maximum modulus principle and the triangle inequality that $|N(M_n)(z) - M_n(z)| \le 2\|f - M_n\|_{\mathbb{T}}$ for $z \in \mathbb{D}$. Since $\exp\{(1 - \varepsilon)m_K\} > 1$, relation (2.19) implies that

$$2\|f - M_n\|_{\mathbb{T}} < \frac{1}{2} \exp \left\{ (1 - \varepsilon) 2n m_K - \frac{2n}{\operatorname{cap}_{\mathbb{D}}(K_f)} \right\}$$

for all $n$ large enough. Hence, by the previous estimates, it holds for all such $n$ and $z \in F_{n, \varepsilon}$ that

$$\left| \frac{N(M_n)(z) - M_n(z)}{f(z) - N(M_n)(z)} \right| < \frac{1}{2} e^{2n(m_K - g(\mu_{\mathbb{D},K_f}, \mathbb{D};z))} \le \frac{1}{2}.$$

In particular, for any $0 < \varepsilon < \varepsilon'$, there exists $n_0$ depending on $K$ and $\varepsilon' - \varepsilon$ such that

$$(3.6) \qquad \frac{1}{2n} \left| \log \left| \frac{f(z) - M_n(z)}{f(z) - N(M_n)(z)} \right| \right| < (\varepsilon' - \varepsilon)m_K$$

for all $z \in F_{n, \varepsilon}$ and $n \ge n_0$. Therefore, we get from the triangle inequality that

$$E(K, \varepsilon' m_K, M_n) \subseteq (K \setminus F_{n, \varepsilon}) \cup E(K, \varepsilon m_K, N(M_n)) = E(K, \varepsilon m_K, N(M_n))$$

for all $n \ge n_0$. The above inclusion clearly yields that (3.5) holds with $N(M_n)$ replaced by $M_n$ for $\varepsilon = \varepsilon' m_K$. As $\varepsilon, \varepsilon'$, and $K$ were arbitrary, the claim follows.

### 3.4. Notation.
We fix $f \in \mathcal{F}(\bar{\mathbb{C}}\setminus\mathbb{D})$ and, with a slight abuse of notation, we keep denoting by $f$ the corresponding function in $\mathcal{F}(\mathcal{R})$ (that was denoted by $\hat{f}$ in (2.8)). Take $\{M_n\}$ to be a sequence of $n$-th root optimal meromorphic approximants of $f$ and let $\{N(M_n)\}$ be the sequence of corresponding Nehari modifications. Since $\{N(M_n)\}$ is also $n$-th root optimal, it holds that

$$(3.7) \qquad \lim_{n \to \infty} \frac{1}{n} \log \|f - N(M_n)\|_{\mathbb{T}} = - \frac{2}{\operatorname{cap}_{\mathbb{D}}(K_f)},$$

see the discussion after (2.19). We shall need an exhaustion of $\mathcal{R}\setminus E_f$ by open sets with “nice” boundaries. That is, we consider a sequence $\{\Omega_m\}_{m \ge 1}$ of open sets such that

$$(3.8) \qquad \Omega_m \subset \mathcal{R}\setminus E_f, \quad \bar{\Omega}_m \subset \Omega_{m+1}, \quad \partial\mathcal{R} \subset \partial\Omega_m, \quad \mathcal{R}\setminus E_f = \cup_m \Omega_m,$$

and each $\Omega_m$, when viewed as an open subset with compact closure of $\mathcal{R}_*$, is regular for the Dirichlet problem, see Section A.7. We will require in addition that $\lambda(\partial\Omega_m \setminus \partial\mathcal{R}) = 0$ for some Radon measure $\lambda$ on $\mathcal{R}$ that will be specified at the beginning of Section 3.5 (a Radon measure is a positive Borel measure which is finite on compact sets). To design regular $\Omega_m$ that meet (3.8) is possible because $E_f$, being compact in $\mathcal{R}$, is a countable intersection of compact sets $K_l \subset \mathcal{R}$ with smooth boundary. Indeed, there is a smooth function $h \ge 0$ on $\mathcal{R}$ such that $E_f$ is the zero set of $h$ and $h \ge c > 0$ outside a compact neighborhood of $E_f$, hence we can pick $K_l$ to be the sublevel set $\{z : h(z) \le t_l\}$, where $\{t_l\}$ a sequence of regular values of $h$ tending to 0 (almost every positive number is a regular value by Sard's theorem). In fact, using smooth partitions of unity and local coordinates, existence of such $h$ quickly reduces to the corresponding issue in Euclidean space, where it follows easily from a combination of [63, Chapter VI, Theorem 2] and [68, Theorem I] (this result is named after H. Whitney). Furthermore, since the sets $C_{a,b} := \{z : a \le h(z) \le b\}$ are compact for $0 < a < b < \varepsilon$ and $\varepsilon > 0$ small enough, $\lambda(C_{a,b}) < \infty$. So, $\lambda(\{z : h(z) = t\}) \ne 0$ for at most countably many positive $t < \varepsilon$, and we can assume that $t_l$ chosen above are not such values. The domains $\Omega_m$ thus constructed satisfy all our requirements.

Let $\Omega$ be a subdomain of $\mathbb{D}$ or $\mathcal{R}$. Given a Borel measure $\sigma$ on $\Omega$, we denote the Green potential of $\sigma$ relative to $\Omega$ by $g(\sigma, \Omega; \cdot)$, see Section A.2. Hereafter, every measure is Borel unless otherwise
and we need to show that (3.5) holds with $N(M_n)$ replaced by $M_n$. Given $\varepsilon \in (0, 1)$, define

$$F_{n, \varepsilon} := \left\{ z \in K : \frac{1}{2n} \log |(f - N(M_n))(z)| > g(\mu_{\mathbb{D},K_f}, \mathbb{D}; \cdot) - \frac{1}{\operatorname{cap}_{\mathbb{D}}(K_f)} - \varepsilon m_K \right\},$$

where $m_K := \min_K g(\mu_{\mathbb{D},K_f}, \mathbb{D}; \cdot) > 0$. Since $N(M_n) - M_n$ is analytic in $\mathbb{D}$, we get from the maximum modulus principle and the triangle inequality that $|N(M_n)(z) - M_n(z)| \le 2\|f - M_n\|_{\mathbb{T}}$ for $z \in \mathbb{D}$. Since $\exp\{(1 - \varepsilon)m_K\} > 1$, relation (2.19) implies that

$$2\|f - M_n\|_{\mathbb{T}} < \frac{1}{2} \exp \left\{ (1 - \varepsilon) 2n m_K - \frac{2n}{\operatorname{cap}_{\mathbb{D}}(K_f)} \right\}$$

for all $n$ large enough. Hence, by the previous estimates, it holds for all such $n$ and $z \in F_{n, \varepsilon}$ that

$$\left| \frac{N(M_n)(z) - M_n(z)}{f(z) - N(M_n)(z)} \right| < \frac{1}{2} e^{2n(m_K - g(\mu_{\mathbb{D},K_f}, \mathbb{D};z))} \le \frac{1}{2}.$$

In particular, for any $0 < \varepsilon < \varepsilon'$, there exists $n_0$ depending on $K$ and $\varepsilon' - \varepsilon$ such that

$$(3.6) \qquad \frac{1}{2n} \left| \log \left| \frac{f(z) - M_n(z)}{f(z) - N(M_n)(z)} \right| \right| < (\varepsilon' - \varepsilon)m_K$$

for all $z \in F_{n, \varepsilon}$ and $n \ge n_0$. Therefore, we get from the triangle inequality that

$$E(K, \varepsilon' m_K, M_n) \subseteq (K \setminus F_{n, \varepsilon}) \cup E(K, \varepsilon m_K, N(M_n)) = E(K, \varepsilon m_K, N(M_n))$$

for all $n \ge n_0$. The above inclusion clearly yields that (3.5) holds with $N(M_n)$ replaced by $M_n$ for $\varepsilon = \varepsilon' m_K$. As $\varepsilon, \varepsilon'$, and $K$ were arbitrary, the claim follows.

### 3.4. Notation.
We fix $f \in \mathcal{F}(\bar{\mathbb{C}}\setminus\mathbb{D})$ and, with a slight abuse of notation, we keep denoting by $f$ the corresponding function in $\mathcal{F}(\mathcal{R})$ (that was denoted by $\hat{f}$ in (2.8)). Take $\{M_n\}$ to be a sequence of $n$-th root optimal meromorphic approximants of $f$ and let $\{N(M_n)\}$ be the sequence of corresponding Nehari modifications. Since $\{N(M_n)\}$ is also $n$-th root optimal, it holds that

$$(3.7) \qquad \lim_{n \to \infty} \frac{1}{n} \log \|f - N(M_n)\|_{\mathbb{T}} = - \frac{2}{\operatorname{cap}_{\mathbb{D}}(K_f)},$$

see the discussion after (2.19). We shall need an exhaustion of $\mathcal{R}\setminus E_f$ by open sets with “nice” boundaries. That is, we consider a sequence $\{\Omega_m\}_{m \ge 1}$ of open sets such that

$$(3.8) \qquad \Omega_m \subset \mathcal{R}\setminus E_f, \quad \bar{\Omega}_m \subset \Omega_{m+1}, \quad \partial\mathcal{R} \subset \partial\Omega_m, \quad \mathcal{R}\setminus E_f = \cup_m \Omega_m,$$

and each $\Omega_m$, when viewed as an open subset with compact closure of $\mathcal{R}_*$, is regular for the Dirichlet problem, see Section A.7. We will require in addition that $\lambda(\partial\Omega_m \setminus \partial\mathcal{R}) = 0$ for some Radon measure $\lambda$ on $\mathcal{R}$ that will be specified at the beginning of Section 3.5 (a Radon measure is a positive Borel measure which is finite on compact sets). To design regular $\Omega_m$ that meet (3.8) is possible because $E_f$, being compact in $\mathcal{R}$, is a countable intersection of compact sets $K_l \subset \mathcal{R}$ with smooth boundary. Indeed, there is a smooth function $h \ge 0$ on $\mathcal{R}$ such that $E_f$ is the zero set of $h$ and $h \ge c > 0$ outside a compact neighborhood of $E_f$, hence we can pick $K_l$ to be the sublevel set $\{z : h(z) \le t_l\}$, where $\{t_l\}$ a sequence of regular values of $h$ tending to 0 (almost every positive number is a regular value by Sard's theorem). In fact, using smooth partitions of unity and local coordinates, existence of such $h$ quickly reduces to the corresponding issue in Euclidean space, where it follows easily from a combination of [63, Chapter VI, Theorem 2] and [68, Theorem I] (this result is named after H. Whitney). Furthermore, since the sets $C_{a,b} := \{z : a \le h(z) \le b\}$ are compact for $0 < a < b < \varepsilon$ and $\varepsilon > 0$ small enough, $\lambda(C_{a,b}) < \infty$. So, $\lambda(\{z : h(z) = t\}) \ne 0$ for at most countably many positive $t < \varepsilon$, and we can assume that $t_l$ chosen above are not such values. The domains $\Omega_m$ thus constructed satisfy all our requirements.

Let $\Omega$ be a subdomain of $\mathbb{D}$ or $\mathcal{R}$. Given a Borel measure $\sigma$ on $\Omega$, we denote the Green potential of $\sigma$ relative to $\Omega$ by $g(\sigma, \Omega; \cdot)$, see Section A.2. Hereafter, every measure is Borel unless otherwise
stated. If $\sigma$ is a measure on a Borel set containing $\Omega$, we write for simplicity $g(\sigma, \Omega; \cdot)$ to mean $g(\sigma|_{\Omega}, \Omega; \cdot)$. Conversely, for a measure $\sigma$ on a Borel set $B_0 \subset \Omega$, we still denote by $\sigma$ the measure on $\Omega$ mapping a Borel set $B$ to $\sigma(B \cap B_0)$, and write $g(\sigma, \Omega; \cdot)$ for its potential. When $\sigma = \sum_j \delta_{z_j}$ is a (possibly infinite) sum of Dirac delta measures, we put

$$ (3.9) \qquad b(\sigma, \Omega; z) = \exp\{-g^*(\sigma, \Omega; z)\} $$

to stand for the corresponding generalized Blaschke product, where $g^*(\sigma, \Omega; \cdot)$ is a complexified Green potential, i.e., it is locally holomorphic in $\Omega\setminus\text{supp}(\sigma)$ and $\text{Re}g^*(\sigma, \Omega; \cdot) = g(\sigma, \Omega; \cdot)$. If $g(\sigma, \Omega; \cdot) \equiv +\infty$, which can happen when the points $z_j$ accumulate in $\Omega$ or to the boundary $\partial\Omega$ sufficiently slowly, then $b(\sigma, \Omega; \cdot)$ is identically zero. Otherwise, $b(\sigma, \Omega; \cdot)$ is well defined up to a unimodular constant (because the periods of a conjugate function of $g(\sigma, \Omega; \cdot)$ are integral multiples of $2\pi$ by Gauss' theorem), holomorphic in $\Omega$, unimodular quasi everywhere on $\partial\Omega$ (everywhere if the latter is regular and the points $z_j$ are finite in number), and it vanishes only at the points $z_j$ (with multiplicities represented by repetition).

**3.5. Stripped Error of Approximation.** We shall study the asymptotics of the error functions $|f - N(M_n) \circ p|$. In this section, we strip off their poles and zeros to take logarithms and obtain harmonic functions whose limiting behavior we then investigate. To this end, we set

$$ (3.10) \qquad \tilde{\mu}_n := \sum_{v_{n,j}} \delta_{v_{n,j}}, \quad \mu_n := \tilde{\mu}_n/n, \quad \text{and} \quad \tilde{v}_n := \sum_{u_{n,j}} \delta_{u_{n,j}}, \quad v_n := \tilde{v}_n/n, $$

where $\{v_{n,j}\} \subset \mathbb{D}$ are the poles of $N(M_n)$ and $\{u_{n,j}\} \subset \mathbb{R}\setminus E_f$ the zeros of $f - N(M_n) \circ p$, with multiplicities counted by repetition. Before we proceed, let us specify the measure $\lambda$ appearing in the definition of the exhaustion $\{\Omega_m\}$ in the previous subsection. To this end, recall that on a locally compact space $X$, a sequence of Radon measures $\{\sigma_n\}$ converges vaguely to a Radon measure $\sigma$ if $\int g d\sigma_n \to \int g d\sigma$ for every $g$ in $C_c(X)$, the space of continuous functions with compact support on $X$ endowed with the sup-norm. Moreover, if the measures $\sigma_n$ are locally bounded on $X$, then they do contain a vaguely convergent subsequence, see for example [14, Theorem 1.41] for an argument on $\mathbb{R}^n$ which is applicable to any $\sigma$-compact locally convex space<fnref n="2" />. Hence, since the measures $\mu_n$ have mass at most 1, we get if $v_n$ is locally bounded along some sequence of integers that there exists a subsequence $N \subset \mathbb{N}$, a measure $v^*$ on $\mathbb{R}\setminus E_f$ and a measure $\mu$ on $\mathbb{D}$ such that $v_n$ and $\mu_n$ converge vaguely to $v^*$ and $\mu$, respectively, along $N$; if the measures $v_n$ have no locally bounded subsequence, i.e., if there exists a compact set $K \subset \mathbb{R}\setminus E_f$ such that $v_n(K) \to \infty$ as $N \ni n \to \infty$, then we put $v^* = 0$ and we only require the vague convergence $\mu_n \to \mu$ along $N$. In any case we take $\lambda$ to be $v^* + \hat{\mu}$, where $\hat{\mu}$ is the lift of $\mu$ to $\mathbb{R}$ defined via (A.33).

Using (3.9), we define Blaschke products vanishing at the poles of $N(M_n)$ and the zeros of $f - N(M_n) \circ p$. Namely, we put

$$ b_n^{pole}(z) := b(\tilde{\mu}_n, \mathbb{D}; z) \quad \text{and} \quad b_{n,m}^{zero}(z) := b(\tilde{v}_n, \Omega_m; z). $$

These functions are not identically zero, as the number of poles of $N(M_n)$ is at most $n$ while the number of zeros of $f - N(M_n) \circ p$ in each $\Omega_m$ is finite. To see the latter point, recall from (3.4) that $|f - N(M_n)|$ is constant on $\mathbb{T}$, hence the error $f - N(M_n)$ can be meromorphically continued across $\mathbb{T}$ by reflection. It implies that $N(M_n)$ can be meromorphically continued across $\mathbb{T}$ and this continuation is necessarily analytic in some neighborhood of $\mathbb{T}$. Thus, $f - N(M_n) \circ p$ is analytic in a neighborhood of $\partial\mathbb{R}$ by the analyticity of $f$ there, so the zeros of $f - N(M_n) \circ p$ can only accumulate on $E_f$ and not on $\partial\mathbb{R}$. Thus, there are at most finitely many of them in each $\Omega_m$.

Using these Blaschke products, we define

$$ (3.11) \qquad h_{n,m}(z) := \frac{1}{n} \log \left| (f - N(M_n) \circ p)(z) \left( \frac{b_n^{pole} \circ p(z)}{b_{n,m}^{zero}(z)} \right) \right|, \quad z \in \Omega_m, $$

<footnote n="2">²Vague convergence is called weak convergence in [14].</footnote>
stated. If $\sigma$ is a measure on a Borel set containing $\Omega$, we write for simplicity $g(\sigma, \Omega; \cdot)$ to mean $g(\sigma|_{\Omega}, \Omega; \cdot)$. Conversely, for a measure $\sigma$ on a Borel set $B_0 \subset \Omega$, we still denote by $\sigma$ the measure on $\Omega$ mapping a Borel set $B$ to $\sigma(B \cap B_0)$, and write $g(\sigma, \Omega; \cdot)$ for its potential. When $\sigma = \sum_j \delta_{z_j}$ is a (possibly infinite) sum of Dirac delta measures, we put

$$ (3.9) \qquad b(\sigma, \Omega; z) = \exp\{-g^*(\sigma, \Omega; z)\} $$

to stand for the corresponding generalized Blaschke product, where $g^*(\sigma, \Omega; \cdot)$ is a complexified Green potential, i.e., it is locally holomorphic in $\Omega\setminus\text{supp}(\sigma)$ and $\text{Re}\,g^*(\sigma, \Omega; \cdot) = g(\sigma, \Omega; \cdot)$. If $g(\sigma, \Omega; \cdot) \equiv +\infty$, which can happen when the points $z_j$ accumulate in $\Omega$ or to the boundary $\partial\Omega$ sufficiently slowly, then $b(\sigma, \Omega; \cdot)$ is identically zero. Otherwise, $b(\sigma, \Omega; \cdot)$ is well defined up to a unimodular constant (because the periods of a conjugate function of $g(\sigma, \Omega; \cdot)$ are integral multiples of $2\pi$ by Gauss' theorem), holomorphic in $\Omega$, unimodular quasi everywhere on $\partial\Omega$ (everywhere if the latter is regular and the points $z_j$ are finite in number), and it vanishes only at the points $z_j$ (with multiplicities represented by repetition).

**3.5. Stripped Error of Approximation.** We shall study the asymptotics of the error functions $|f - N(M_n) \circ p|$. In this section, we strip off their poles and zeros to take logarithms and obtain harmonic functions whose limiting behavior we then investigate. To this end, we set

$$ (3.10) \qquad \tilde{\mu}_n := \sum_{v_{n,j}} \delta_{v_{n,j}}, \quad \mu_n := \tilde{\mu}_n/n, \quad \text{and} \quad \tilde{v}_n := \sum_{u_{n,j}} \delta_{u_{n,j}}, \quad v_n := \tilde{v}_n/n, $$

where $\{v_{n,j}\} \subset \mathbb{D}$ are the poles of $N(M_n)$ and $\{u_{n,j}\} \subset \mathbb{R}\setminus E_f$ the zeros of $f - N(M_n) \circ p$, with multiplicities counted by repetition. Before we proceed, let us specify the measure $\lambda$ appearing in the definition of the exhaustion $\{\Omega_m\}$ in the previous subsection. To this end, recall that on a locally compact space $X$, a sequence of Radon measures $\{\sigma_n\}$ converges vaguely to a Radon measure $\sigma$ if $\int g d\sigma_n \to \int g d\sigma$ for every $g$ in $C_c(X)$, the space of continuous functions with compact support on $X$ endowed with the sup-norm. Moreover, if the measures $\sigma_n$ are locally bounded on $X$, then they do contain a vaguely convergent subsequence, see for example [14, Theorem 1.41] for an argument on $\mathbb{R}^n$ which is applicable to any $\sigma$-compact locally convex space<fnref n="2" />. Hence, since the measures $\mu_n$ have mass at most 1, we get if $v_n$ is locally bounded along some sequence of integers that there exists a subsequence $N \subset \mathbb{N}$, a measure $v^*$ on $\mathbb{R}\setminus E_f$ and a measure $\mu$ on $\mathbb{D}$ such that $v_n$ and $\mu_n$ converge vaguely to $v^*$ and $\mu$, respectively, along $N$; if the measures $v_n$ have no locally bounded subsequence, i.e., if there exists a compact set $K \subset \mathbb{R}\setminus E_f$ such that $v_n(K) \to \infty$ as $N \ni n \to \infty$, then we put $v^* = 0$ and we only require the vague convergence $\mu_n \to \mu$ along $N$. In any case we take $\lambda$ to be $v^* + \hat{\mu}$, where $\hat{\mu}$ is the lift of $\mu$ to $\mathbb{R}$ defined via (A.33).

Using (3.9), we define Blaschke products vanishing at the poles of $N(M_n)$ and the zeros of $f - N(M_n) \circ p$. Namely, we put

$$ b_n^{\text{pole}}(z) := b(\tilde{\mu}_n, \mathbb{D}; z) \quad \text{and} \quad b_{n,m}^{\text{zero}}(z) := b(\tilde{v}_n, \Omega_m; z). $$

These functions are not identically zero, as the number of poles of $N(M_n)$ is at most $n$ while the number of zeros of $f - N(M_n) \circ p$ in each $\Omega_m$ is finite. To see the latter point, recall from (3.4) that $|f - N(M_n)|$ is constant on $\mathbb{T}$, hence the error $f - N(M_n)$ can be meromorphically continued across $\mathbb{T}$ by reflection. It implies that $N(M_n)$ can be meromorphically continued across $\mathbb{T}$ and this continuation is necessarily analytic in some neighborhood of $\mathbb{T}$. Thus, $f - N(M_n) \circ p$ is analytic in a neighborhood of $\partial\mathbb{R}$ by the analyticity of $f$ there, so the zeros of $f - N(M_n) \circ p$ can only accumulate on $E_f$ and not on $\partial\mathbb{R}$. Thus, there are at most finitely many of them in each $\Omega_m$.

Using these Blaschke products, we define

$$ (3.11) \qquad h_{n,m}(z) := \frac{1}{n} \log \left| (f - N(M_n) \circ p)(z) \left( \frac{b_n^{\text{pole}} \circ p(z)}{b_{n,m}^{\text{zero}}(z)} \right) \right|, \quad z \in \Omega_m, $$

<footnote n="2">²Vague convergence is called weak convergence in [14].</footnote>
which is harmonic in $\Omega_m$. Recall that superharmonic functions on a hyperbolic Riemann surface are either identically $+\infty$ or finite quasi everywhere, and any two of them that coincide almost everywhere (with respect to Lebesgue measure in local coordinates) are in fact equal (the weak identity principle), see Section A.1.

**Lemma 3.1.** There exist a subsequence $N' \subseteq N$ and a non-negative superharmonic function $u'(z)$ on $\mathbb{R}$ such that

$$ (3.12) \qquad -h_{n,m}(z) \to u'(z) \quad \text{as} \quad m \to \infty, $$

locally uniformly in $z \in \mathbb{R} \setminus E_f$, for any subsequence $\{n_m\}_{m=1}^\infty \subseteq N'$. If $u'$ is finite quasi everywhere, then one has a decomposition

$$ (3.13) \qquad u' = g(v', \mathcal{R}; \cdot) + h', $$

where $v'$ is a finite positive Borel measure supported on $E_f$ and $h'$ is a non-negative harmonic function on $\mathbb{R}$.

*Proof.* The regularity of $\partial\Omega_m$ implies that $|b_{n,m}^{\text{zero}}(z)| \equiv 1$ for $z \in \partial\Omega_m$, and likewise $|b_n^{\text{pole}}(z)| = 1$ for $z$ on $\mathbb{T}$. In particular we get that

$$ |N(M_n)(z)b_n^{\text{pole}}(z)| \le \|N(M_n)\|_{\mathbb{T}}, \quad z \in \mathbb{D}, $$

by the maximum modulus principle for $H^\infty$-functions. Set $\Gamma_n := \Gamma_{f-M_n}$ be the Hankel operator with symbol $f - M_n$, see (2.23). It follows from the definition of the singular values, (2.24), and (3.3) that $\|\Gamma_n\| = s_0(\Gamma_n) = \|f - N(M_n)\|_{\mathbb{T}} \le \|f - M_n\|_{\mathbb{T}}$. Since the norms $\|f - M_n\|_{\mathbb{T}}$ tend to zero by assumption, we get that

$$ (3.14) \qquad \|N(M_n)\|_{\mathbb{T}} \le \|f\|_{\mathbb{T}} + \|\Gamma_n\| \le C_f $$

for some constant $C_f$ that depends only on $f$ and the sequence $\{M_n\}$. Thus, we get from the maximum principle for harmonic functions that

$$ (3.15) \qquad h_{n,m}(z) \le \frac{1}{n} \log (\|f\|_{\partial\Omega_m} + C_f), \quad z \in \Omega_m. $$

Set $N_0 := N$. Proceeding inductively on $m \ge 1$, we deduce from (3.15) that the sequence $\{h_{n,m}\}_{n \in N_{m-1}}$ is uniformly bounded above in $\Omega_m$ and therefore, by Harnack's theorem, see Section A.1, there exists a subsequence $N_m \subseteq N_{m-1}$ such that

$$ (3.16) \qquad h_{n,m}(z) \to h_m(z) \quad \text{as} \quad N_m \ni n \to \infty $$

locally uniformly in $\Omega_m$, where $h_m(z)$ is either identically $-\infty$ or a non-positive harmonic function. Define $N_*$ to be the diagonal of the table $\{N_m\}_{m=1}^\infty$; that is, the $m$-th element of $N_*$ is the $m$-th element of $N_m$. Then, (3.16) holds along $n \in N_*^\infty$ for each $m$.

Additionally, it follows from the maximum modulus principle for holomorphic functions that

$$ |b_{n,m_1}^{\text{zero}}(z)| > |b_{n,m_2}^{\text{zero}}(z)|, \quad z \in \Omega_{m_1}, \quad m_1 < m_2. $$

Therefore, we get from (3.11) and (3.15) that $h_{m_1}(z) \le h_{m_2}(z) \le 0$ for $z \in \Omega_{m_1}$ when $m_1 < m_2$. Thus, if a finite limit $h_{m_*}(z)$ exists for some index $m_*$, then it exists for all $m > m_*$. Hence, either the functions $h_{n,m}$ converge to $-\infty$ as $N_* \ni n \to \infty$ locally uniformly in each $\Omega_m$, in which case $h_m \equiv -\infty$ for all $m$, or else the functions $h_m$ are finite and harmonic for all $m$ large enough.

If $h_m \equiv -\infty$ for all $m$, select for each $m$ some $n_m \in N_*$ such that $h_{n,m}(z) < -m$ for $z \in \bar{\Omega}_{m-1}$ and all $n \ge n_m$; we may require in addition that $n_m > n_{m-1}$ for $m \ge 1$. Then (3.12) holds with $N' := \{n_m\}_{m=1}^\infty$ and $u' \equiv +\infty$.

If on the contrary the functions $h_m$ are finite, they form an increasing sequence on $\Omega_\ell$ for $m \ge \ell$ and fixed $\ell$. As they are non-positive, they converge locally uniformly in $\mathbb{R} \setminus E_f$ to a non-positive harmonic function, say $-u'$, again by Harnack's theorem. Since $E_f$ is a closed polar set and $u'$ is non-negative, it follows from the Removability theorem, see Section A.5, that $u'$ extends to a
which is harmonic in $\Omega_m$. Recall that superharmonic functions on a hyperbolic Riemann surface are either identically $+\infty$ or finite quasi everywhere, and any two of them that coincide almost everywhere (with respect to Lebesgue measure in local coordinates) are in fact equal (the weak identity principle), see Section A.1.

**Lemma 3.1.** There exist a subsequence $N' \subseteq N$ and a non-negative superharmonic function $u'(z)$ on $\mathbb{R}$ such that

$$ (3.12) \qquad -h_{n,m}(z) \to u'(z) \quad \text{as} \quad m \to \infty, $$

locally uniformly in $z \in \mathbb{R} \setminus E_f$, for any subsequence $\{n_m\}_{m=1}^\infty \subseteq N'$. If $u'$ is finite quasi everywhere, then one has a decomposition

$$ (3.13) \qquad u' = g(v', \mathbb{R}; \cdot) + h', $$

where $v'$ is a finite positive Borel measure supported on $E_f$ and $h'$ is a non-negative harmonic function on $\mathbb{R}$.

*Proof.* The regularity of $\partial\Omega_m$ implies that $|b_{n,m}^{\text{zero}}(z)| = 1$ for $z \in \partial\Omega_m$, and likewise $|b_n^{\text{pole}}(z)| = 1$ for $z$ on $\mathbb{T}$. In particular we get that

$$ |N(M_n)(z)b_n^{\text{pole}}(z)| \le \|N(M_n)\|_{\mathbb{T}}, \quad z \in \mathbb{D}, $$

by the maximum modulus principle for $H^\infty$-functions. Set $\Gamma_n := \Gamma_{f-M_n}$ be the Hankel operator with symbol $f - M_n$, see (2.23). It follows from the definition of the singular values, (2.24), and (3.3) that $\|\Gamma_n\| = s_0(\Gamma_n) = \|f - N(M_n)\|_{\mathbb{T}} \le \|f - M_n\|_{\mathbb{T}}$. Since the norms $\|f - M_n\|_{\mathbb{T}}$ tend to zero by assumption, we get that

$$ (3.14) \qquad \|N(M_n)\|_{\mathbb{T}} \le \|f\|_{\mathbb{T}} + \|\Gamma_n\| \le C_f $$

for some constant $C_f$ that depends only on $f$ and the sequence $\{M_n\}$. Thus, we get from the maximum principle for harmonic functions that

$$ (3.15) \qquad h_{n,m}(z) \le \frac{1}{n} \log (\|f\|_{\partial\Omega_m} + C_f), \quad z \in \Omega_m. $$

Set $N_0 := N$. Proceeding inductively on $m \ge 1$, we deduce from (3.15) that the sequence $\{h_{n,m}\}_{n \in N_{m-1}}$ is uniformly bounded above in $\Omega_m$ and therefore, by Harnack's theorem, see Section A.1, there exists a subsequence $N_m \subseteq N_{m-1}$ such that

$$ (3.16) \qquad h_{n,m}(z) \to h_m(z) \quad \text{as} \quad N_m \ni n \to \infty $$

locally uniformly in $\Omega_m$, where $h_m(z)$ is either identically $-\infty$ or a non-positive harmonic function. Define $N_*$ to be the diagonal of the table $\{N_m\}_{m=1}^\infty$; that is, the $m$-th element of $N_*$ is the $m$-th element of $N_m$. Then, (3.16) holds along $n \in N_*$ for each $m$.

Additionally, it follows from the maximum modulus principle for holomorphic functions that

$$ |b_{n,m_1}^{\text{zero}}(z)| > |b_{n,m_2}^{\text{zero}}(z)|, \quad z \in \Omega_{m_1}, \quad m_1 < m_2. $$

Therefore, we get from (3.11) and (3.15) that $h_{m_1}(z) \le h_{m_2}(z) \le 0$ for $z \in \Omega_{m_1}$ when $m_1 < m_2$. Thus, if a finite limit $h_{m_*}(z)$ exists for some index $m_*$, then it exists for all $m > m_*$. Hence, either the functions $h_{n,m}$ converge to $-\infty$ as $N_* \ni n \to \infty$ locally uniformly in each $\Omega_m$, in which case $h_m \equiv -\infty$ for all $m$, or else the functions $h_m$ are finite and harmonic for all $m$ large enough.

If $h_m \equiv -\infty$ for all $m$, select for each $m$ some $n_m \in N_*$ such that $h_{n,m}(z) < -m$ for $z \in \bar{\Omega}_{m-1}$ and all $n \ge n_m$; we may require in addition that $n_m > n_{m-1}$ for $m \ge 1$. Then (3.12) holds with $N' := \{n_m\}_{m=1}^\infty$ and $u' \equiv +\infty$.

If on the contrary the functions $h_m$ are finite, they form an increasing sequence on $\Omega_\ell$ for $m \ge \ell$ and fixed $\ell$. As they are non-positive, they converge locally uniformly in $\mathbb{R} \setminus E_f$ to a non-positive harmonic function, say $-u'$, again by Harnack's theorem. Since $E_f$ is a closed polar set and $u'$ is non-negative, it follows from the Removability theorem, see Section A.5, that $u'$ extends to a
superharmonic function on $\mathbb{R}$ that we keep denoting by $u'$. Because $u'$ is superharmonic on $\mathbb{R}$ and harmonic on $\mathbb{R}\setminus E_f$, its Laplacian is a negative Radon measure $-v'$ supported on $E_f$, which is necessarily finite since $E_f$ is compact. Thus, by the Riesz representation theorem, see Section A.3, equation (3.13) takes place with $h'$ the largest harmonic minorant of $u'$.

Given $m$, choose $\tilde{n}_m \in \mathcal{N}_*$ such that $|h_{n,m}(z) - h_m(z)| \le 1/m$ for $z \in \bar{\Omega}_{m-1}$ and all $n \ge \tilde{n}_m$. Define $\mathcal{N}' := \{\tilde{n}_m\}_{m=1}^\infty$, where we again additionally require that $\tilde{n}_m > \tilde{n}_{m-1}$. Given a compact set $K \subset \mathbb{R}\setminus E_f$ and $\epsilon > 0$, we can pick $m$ large enough that

$$K \subseteq \bar{\Omega}_{m-1}, \quad 1/m \le \epsilon/2, \quad \text{and} \quad |u'(z) + h_m(z)| \le \epsilon/2, \quad z \in K.$$

Then, it follows from the last two inequalities that $|h_{n,m}(z) + u'(z)| \le \epsilon$ for $z \in K$ and any $\mathcal{N}' \ni n \ge \tilde{n}_m$. Since $\epsilon$ and $K$ were arbitrary, this finishes the proof of (3.12). $\square$

**Lemma 3.2.** If $u' \ne +\infty$ in Lemma 3.1, then $h'$ in (3.13) continuously extends to $\partial\mathbb{R}$ and

$$(3.17) \qquad h'(z) = \begin{cases} \frac{2}{\operatorname{cap}_{\mathbb{D}}(K_f)}, & z \in \mathcal{T}, \\ 0, & z \in \partial\mathbb{R}\setminus\mathcal{T}. \end{cases}$$

*Proof.* Let $\Omega_* := p^{-1}(\{z : r < |z| < 1\})$, with $r > 0$ close enough to 1 that $\bar{\Omega}_*\setminus\partial\mathbb{R} \subset \Omega_m$ for each $m$. It follows from the proof of Lemma 3.1 that $h_{n,m}(z) \to h_m(z)$ as $\mathcal{N}' \ni n \to \infty$, locally uniformly in $\bar{\Omega}_*\setminus\partial\mathbb{R}$. Given a connected component $\Omega$ of $\Omega_*$, let $\delta_z^{\mathbb{R}\setminus\Omega}$ be its harmonic measure, see Section A.9. Then

$$(3.18) \qquad \log|p(z) - (1 + \epsilon)p(\xi)| = \int \log|p(\zeta) - (1 + \epsilon)p(\xi)|d\delta_z^{\mathbb{R}\setminus\Omega}(\zeta), \quad z \in \Omega,$$

for any $\xi \in \partial\Omega \cap \partial\mathbb{R}$ and $\epsilon > 0$, see (A.28). It then follows from the monotone convergence theorem that we can take $\epsilon = 0$ in (3.18). Recall that $f - N(M_n) \circ p$ is analytic across $\partial\mathbb{R} \cap \partial\Omega$ and hence is non-vanishing there except perhaps for finitely many zeros counting multiplicities. Since $h_{n,m}$ is harmonic in $\Omega$, is continuous on $\partial\Omega\setminus\partial\mathbb{R}$, and is equal to $\frac{1}{n}\log|f - N(M_n) \circ p|$ on $\partial\mathbb{R} \cap \partial\Omega$ (i.e., it is continuous on $\partial\Omega\setminus\partial\mathbb{R}$ except perhaps for finitely many logarithmic singularities), we get from (3.18) and (A.28) that

$$h_{n,m}(z) = \int_{\partial\Omega\setminus\partial\mathbb{R}} h_{n,m} d\delta_z^{\mathbb{R}\setminus\Omega} + \int_{\partial\mathbb{R} \cap \partial\Omega} \frac{1}{n} \log|f - N(M_n) \circ p| d\delta_z^{\mathbb{R}\setminus\Omega}, \quad z \in \Omega.$$

By (3.7) the pointwise limit of $\frac{1}{n}\log|f - N(M_n) \circ p|$ on $\partial\mathbb{R}$ is minus the right-hand side of (3.17) except perhaps for a finite subset of $\partial\mathbb{R}\setminus\mathcal{T}$, where $|f - N(M_n) \circ p|$ may go to zero, contained in

$$\{\zeta | \exists \eta \ne \zeta : p(\zeta) = p(\eta), f(\zeta) = f(\eta)\} \setminus \mathcal{T}.$$

As $\delta_z^{\mathbb{R}\setminus\Omega}$ does not charge polar, thus finite sets, the convergence in fact holds almost everywhere with respect to $\delta_z^{\mathbb{R}\setminus\Omega}$ for each fixed $z$. So, if we can justify the second equality in the following relation:

$$(3.19) \qquad h_m(z) = \lim_{\mathcal{N}' \ni n \to \infty} \left( \int_{\partial\Omega \setminus \partial\mathbb{R}} h_{n,m} d\delta_z^{\mathbb{R}\setminus\Omega} + \int_{\partial\mathbb{R} \cap \partial\Omega} \frac{1}{n} \log|f - N(M_n) \circ p| d\delta_z^{\mathbb{R}\setminus\Omega} \right) \\ = \int_{\partial\Omega \setminus \partial\mathbb{R}} h_m d\delta_z^{\mathbb{R}\setminus\Omega} + \int_{\partial\mathbb{R} \cap \partial\Omega} \lim_{N \ni n \to \infty} \frac{1}{n} \log|f - N(M_n) \circ p| d\delta_z^{\mathbb{R}\setminus\Omega},$$

we shall get that $h_m(z)$ solves a Dirichlet problem on $\Omega$ with boundary data equal to $h_{m|\partial\Omega\setminus\partial\mathbb{R}}$ on $\partial\Omega\setminus\partial\mathbb{R}$, and to the negative of the right-hand side of (3.17) on $\partial\mathbb{R} \cap \partial\Omega$. As such, $h_m$ must extend continuously to $\partial\Omega$ where it is equal to the boundary data, since $\partial\Omega$ is non-thin at any of its points. Subsequently, as $h_m$ converges to $-u'$ locally uniformly on $\mathbb{R}\setminus E_f$ (see the proof of Lemma 3.1), passing to the limit in the leftmost and rightmost sides of (3.19) when $m \to \infty$ yields that $u'$ extends continuously to $\partial\Omega \cap \partial\mathbb{R}$ with values given there by the right-hand side of (3.17). This will give us
superharmonic function on $\mathbb{R}$ that we keep denoting by $u'$. Because $u'$ is superharmonic on $\mathbb{R}$ and harmonic on $\mathbb{R}\setminus E_f$, its Laplacian is a negative Radon measure $-v'$ supported on $E_f$, which is necessarily finite since $E_f$ is compact. Thus, by the Riesz representation theorem, see Section A.3, equation (3.13) takes place with $h'$ the largest harmonic minorant of $u'$.

Given $m$, choose $\tilde{n}_m \in \mathcal{N}_*$ such that $|h_{n,m}(z) - h_m(z)| \le 1/m$ for $z \in \bar{\Omega}_{m-1}$ and all $n \ge \tilde{n}_m$. Define $\mathcal{N}' := \{\tilde{n}_m\}_{m=1}^\infty$, where we again additionally require that $\tilde{n}_m > \tilde{n}_{m-1}$. Given a compact set $K \subset \mathbb{R}\setminus E_f$ and $\epsilon > 0$, we can pick $m$ large enough that

$$K \subseteq \bar{\Omega}_{m-1}, \quad 1/m \le \epsilon/2, \quad \text{and} \quad |u'(z) + h_m(z)| \le \epsilon/2, \quad z \in K.$$

Then, it follows from the last two inequalities that $|h_{n,m}(z) + u'(z)| \le \epsilon$ for $z \in K$ and any $\mathcal{N}' \ni n \ge \tilde{n}_m$. Since $\epsilon$ and $K$ were arbitrary, this finishes the proof of (3.12). $\square$

**Lemma 3.2.** If $u' \ne +\infty$ in Lemma 3.1, then $h'$ in (3.13) continuously extends to $\partial\mathbb{R}$ and

$$(3.17) \qquad h'(z) = \begin{cases} \frac{2}{\operatorname{cap}_{\mathbb{D}}(K_f)}, & z \in \mathcal{T}, \\ 0, & z \in \partial\mathbb{R}\setminus\mathcal{T}. \end{cases}$$

*Proof.* Let $\Omega_* := p^{-1}(\{z : r < |z| < 1\})$, with $r > 0$ close enough to 1 that $\bar{\Omega}_*\setminus\partial\mathbb{R} \subset \Omega_m$ for each $m$. It follows from the proof of Lemma 3.1 that $h_{n,m}(z) \to h_m(z)$ as $\mathcal{N}' \ni n \to \infty$, locally uniformly in $\bar{\Omega}_*\setminus\partial\mathbb{R}$. Given a connected component $\Omega$ of $\Omega_*$, let $\delta_z^{\mathbb{R}\setminus\Omega}$ be its harmonic measure, see Section A.9. Then

$$(3.18) \qquad \log|p(z) - (1 + \epsilon)p(\xi)| = \int \log|p(\zeta) - (1 + \epsilon)p(\xi)|d\delta_z^{\mathbb{R}\setminus\Omega}(\zeta), \quad z \in \Omega,$$

for any $\xi \in \partial\Omega \cap \partial\mathbb{R}$ and $\epsilon > 0$, see (A.28). It then follows from the monotone convergence theorem that we can take $\epsilon = 0$ in (3.18). Recall that $f - N(M_n) \circ p$ is analytic across $\partial\mathbb{R} \cap \partial\Omega$ and hence is non-vanishing there except perhaps for finitely many zeros counting multiplicities. Since $h_{n,m}$ is harmonic in $\Omega$, is continuous on $\partial\Omega\setminus\partial\mathbb{R}$, and is equal to $\frac{1}{n}\log|f - N(M_n) \circ p|$ on $\partial\mathbb{R} \cap \partial\Omega$ (i.e., it is continuous on $\partial\Omega\setminus\partial\mathbb{R}$ except perhaps for finitely many logarithmic singularities), we get from (3.18) and (A.28) that

$$h_{n,m}(z) = \int_{\partial\Omega\setminus\partial\mathbb{R}} h_{n,m} d\delta_z^{\mathbb{R}\setminus\Omega} + \int_{\partial\mathbb{R} \cap \partial\Omega} \frac{1}{n} \log|f - N(M_n) \circ p| d\delta_z^{\mathbb{R}\setminus\Omega}, \quad z \in \Omega.$$

By (3.7) the pointwise limit of $\frac{1}{n}\log|f - N(M_n) \circ p|$ on $\partial\mathbb{R}$ is minus the right-hand side of (3.17) except perhaps for a finite subset of $\partial\mathbb{R}\setminus\mathcal{T}$, where $|f - N(M_n) \circ p|$ may go to zero, contained in

$$\{\zeta | \exists \eta \ne \zeta : p(\zeta) = p(\eta), f(\zeta) = f(\eta)\} \setminus \mathcal{T}.$$

As $\delta_z^{\mathbb{R}\setminus\Omega}$ does not charge polar, thus finite sets, the convergence in fact holds almost everywhere with respect to $\delta_z^{\mathbb{R}\setminus\Omega}$ for each fixed $z$. So, if we can justify the second equality in the following relation:

$$(3.19) \qquad h_m(z) = \lim_{\mathcal{N}' \ni n \to \infty} \left( \int_{\partial\Omega \setminus \partial\mathbb{R}} h_{n,m} d\delta_z^{\mathbb{R}\setminus\Omega} + \int_{\partial\mathbb{R} \cap \partial\Omega} \frac{1}{n} \log|f - N(M_n) \circ p| d\delta_z^{\mathbb{R}\setminus\Omega} \right) \\ = \int_{\partial\Omega \setminus \partial\mathbb{R}} h_m d\delta_z^{\mathbb{R}\setminus\Omega} + \int_{\partial\mathbb{R} \cap \partial\Omega} \lim_{N \ni n \to \infty} \frac{1}{n} \log|f - N(M_n) \circ p| d\delta_z^{\mathbb{R}\setminus\Omega},$$

we shall get that $h_m(z)$ solves a Dirichlet problem on $\Omega$ with boundary data equal to $h_{m|\partial\Omega\setminus\partial\mathbb{R}}$ on $\partial\Omega\setminus\partial\mathbb{R}$, and to the negative of the right-hand side of (3.17) on $\partial\mathbb{R} \cap \partial\Omega$. As such, $h_m$ must extend continuously to $\partial\Omega$ where it is equal to the boundary data, since $\partial\Omega$ is non-thin at any of its points. Subsequently, as $h_m$ converges to $-u'$ locally uniformly on $\mathbb{R}\setminus E_f$ (see the proof of Lemma 3.1), passing to the limit in the leftmost and rightmost sides of (3.19) when $m \to \infty$ yields that $u'$ extends continuously to $\partial\Omega \cap \partial\mathbb{R}$ with values given there by the right-hand side of (3.17). This will give us
the desired conclusion, because $v'$ is compactly supported in $\mathcal{R}$ and therefore $g(v', \mathcal{R}; \cdot)$ continuously extends by zero to $\partial\mathcal{R}$ since $\mathcal{R}$ is regular.

Altogether, it only remains to justify the swapping of the limit and integration signs in (3.19). On $\partial\Omega \setminus \partial\mathcal{R}$, one can invoke the dominated convergence theorem. Thus, we only need to consider the integral over $\partial\mathcal{R} \cap \partial\Omega$. According to Vitali's convergence theorem, it is enough to show that for every $\epsilon > 0$ there exists $\delta > 0$ and $n_\epsilon \in \mathbb{N}$ for which

$$ (3.20) \quad \int_E \left| \frac{1}{n} \log |f - N(M_n) \circ p| \right| d\delta_z^{\mathcal{R}\setminus\Omega} < \epsilon \quad \text{as soon as} \quad |\delta_z^{\mathcal{R}\setminus\Omega}(E)| < \delta \quad \text{and} \quad n \ge n_\epsilon. $$

For this, we first deduce from (3.14) that

$$ (3.21) \quad \int_{\partial\mathcal{R}} \frac{1}{n} \log |f - N(M_n) \circ p| d\delta_z^{\mathcal{R}\setminus\Omega} \le \frac{1}{n} \log (\|f\|_{\partial\mathcal{R}} + C_f) $$

as $\delta_z^{\mathcal{R}\setminus\Omega}$ is a probability measure. Now, if $\partial\mathcal{R} \cap \partial\Omega = \mathcal{T}$, then it follows from (3.7) that

$$ (3.22) \quad \int_E \frac{1}{n} \log |f - N(M_n) \circ p| d\delta_z^{\mathcal{R}\setminus\Omega} > -c\delta_z^{\mathcal{R}\setminus\Omega}(E), $$

for any $E \subseteq \mathcal{T}$ and some positive constant $c$. If $\partial\mathcal{R} \cap \partial\Omega = \mathcal{T}' \ne \mathcal{T}$, set $d(\eta) := f(\eta) - f(\zeta)$ where $\eta \in \mathcal{T}', \zeta \in \mathcal{T}$, and $p(\eta) = p(\zeta)$. Then, we get from (3.14) that

$$ (3.23) \quad \log |(f - N(M_n) \circ p)(\eta)| \ge \log \|d(\eta) - \|\Gamma_n\|\| = \log \left| \frac{|d(\eta)|^2 - \|\Gamma_n\|^2}{|d(\eta)| + \|\Gamma_n\|} \right| \\ \ge \log \|d(\eta)|^2 - \|\Gamma_n\|^2\| - \log(\|f\|_{\mathcal{T}'} + C_f). $$

Since $E_f$, the singular set of $f$, and $\mathbf{rp}(\mathcal{R})$, the ramification set of $\mathcal{R}$, are closed and lie on top of $\mathbb{D}$, $d(\eta)$ extends to a holomorphic function, non-identically zero in a neighborhood of $\mathcal{T}'$. Then,

$$ \mathcal{D}(\eta) := d(\eta)d(p^{-1}(1/p(\eta))) \quad \text{satisfies} \quad \mathcal{D}(\eta) = |d(\eta)|^2, \quad \eta \in \mathcal{T}', $$

and is holomorphic about $\mathcal{T}'$. Pick an open set $W \supset \mathcal{T}'$ such that $\overline{W} \cap \mathbf{rp}(\mathcal{R}) = \emptyset$ and $\mathcal{D}$ is holomorphic in $\overline{W}$ with no zero on $\partial W$; then, so is $\mathcal{D} - \|\Gamma_n\|^2$ for $n$ large as $\|\Gamma_n\|^2 \to 0$. Let $\ell$ and $\ell_n$ be minimal degree polynomials, normalized by imposing $\|\ell\|_{\mathcal{T}} = \|\ell_n\|_{\mathcal{T}} = 1$, such that

$$ \frac{\mathcal{D}(\eta)}{\ell(p(\eta))} \quad \text{and} \quad \frac{\mathcal{D}(\eta) - \|\Gamma_n\|^2}{\ell_n(p(\eta))} $$

are holomorphic and non-vanishing in $W$. Since $\|\Gamma_n\|^2 \to 0$, the zeros of $\mathcal{D} - \|\Gamma_n\|^2$ in $W$ tend to those of $\mathcal{D}$ by Rouché's theorem, and so our normalization implies that $\ell_n \to \ell$ uniformly in $\overline{W}$ as $n \to \infty$. Hence, $(\mathcal{D} - \|\Gamma_n\|^2)/(\ell_n \circ p)$ converges to $\mathcal{D}/(\ell_n \circ p)$ uniformly in $\overline{W}$, in particular it is uniformly bounded away from zero there. Consequently, it follows from (3.23) that

$$ (3.24) \quad \log |(f - N(M_n) \circ p)(\eta)| \ge C + \log |\ell_n(p(\eta))|, \quad \eta \in \mathcal{T}', $$

for some finite constant $C$. Note that $\log |\ell_n| \le 0$ in $\mathbb{D}$ according to our normalization. Let us write $\ell_n(x) = a_n \prod_i (x - x_{i,n})$ and define the reciprocal polynomial $\tilde{\ell}_n$ of $\ell_n$ by

$$ \tilde{\ell}_n(x) := a_n \prod_i \begin{cases} x - x_{i,n} & \text{if } |x_{i,n}| \ge 1, \\ 1 - x \tilde{x}_{i,n} & \text{if } |x_{i,n}| < 1. \end{cases} $$

Clearly, $|\ell_n(\xi)| = |\tilde{\ell}_n(\xi)|$ for $|\xi| = 1$, and the maximum principle for harmonic functions implies that

$$ (3.25) \quad \int_{\mathcal{T}'} \log |\tilde{\ell}_n(p(\eta))| d\delta_z^{\mathcal{R}\setminus\Omega} \ge \log |\tilde{\ell}_n(p(z))|, \quad z \in \Omega, $$
the desired conclusion, because $v'$ is compactly supported in $\mathcal{R}$ and therefore $g(v', \mathcal{R}; \cdot)$ continuously extends by zero to $\partial\mathcal{R}$ since $\mathcal{R}$ is regular.

Altogether, it only remains to justify the swapping of the limit and integration signs in (3.19). On $\partial\Omega \setminus \partial\mathcal{R}$, one can invoke the dominated convergence theorem. Thus, we only need to consider the integral over $\partial\mathcal{R} \cap \partial\Omega$. According to Vitali's convergence theorem, it is enough to show that for every $\epsilon > 0$ there exists $\delta > 0$ and $n_\epsilon \in \mathbb{N}$ for which

$$ (3.20) \quad \int_E \left| \frac{1}{n} \log |f - N(M_n) \circ p| \right| d\delta_z^{\mathcal{R}\setminus\Omega} < \epsilon \quad \text{as soon as} \quad |\delta_z^{\mathcal{R}\setminus\Omega}(E)| < \delta \quad \text{and} \quad n \ge n_\epsilon. $$

For this, we first deduce from (3.14) that

$$ (3.21) \quad \int_{\partial\mathcal{R}} \frac{1}{n} \log |f - N(M_n) \circ p| d\delta_z^{\mathcal{R}\setminus\Omega} \le \frac{1}{n} \log (\|f\|_{\partial\mathcal{R}} + C_f) $$

as $\delta_z^{\mathcal{R}\setminus\Omega}$ is a probability measure. Now, if $\partial\mathcal{R} \cap \partial\Omega = \mathcal{T}$, then it follows from (3.7) that

$$ (3.22) \quad \int_E \frac{1}{n} \log |f - N(M_n) \circ p| d\delta_z^{\mathcal{R}\setminus\Omega} > -c\delta_z^{\mathcal{R}\setminus\Omega}(E), $$

for any $E \subseteq \mathcal{T}$ and some positive constant $c$. If $\partial\mathcal{R} \cap \partial\Omega = \mathcal{T}' \ne \mathcal{T}$, set $d(\eta) := f(\eta) - f(\zeta)$ where $\eta \in \mathcal{T}', \zeta \in \mathcal{T}$, and $p(\eta) = p(\zeta)$. Then, we get from (3.14) that

$$ (3.23) \quad \log |(f - N(M_n) \circ p)(\eta)| \ge \log \|d(\eta) - \|\Gamma_n\|\| = \log \left| \frac{|d(\eta)|^2 - \|\Gamma_n\|^2}{|d(\eta)| + \|\Gamma_n\|} \right| \\ \qquad \ge \log \|d(\eta)|^2 - \|\Gamma_n\|^2 - \log(\|f\|_{\mathcal{T}'} + C_f). $$

Since $E_f$, the singular set of $f$, and $\mathbf{rp}(\mathcal{R})$, the ramification set of $\mathcal{R}$, are closed and lie on top of $\mathbb{D}$, $d(\eta)$ extends to a holomorphic function, non-identically zero in a neighborhood of $\mathcal{T}'$. Then,

$$ \mathcal{D}(\eta) := d(\eta)d(p^{-1}(1/p(\eta))) \quad \text{satisfies} \quad \mathcal{D}(\eta) = |d(\eta)|^2, \quad \eta \in \mathcal{T}', $$

and is holomorphic about $\mathcal{T}'$. Pick an open set $W \supset \mathcal{T}'$ such that $\overline{W} \cap \mathbf{rp}(\mathcal{R}) = \emptyset$ and $\mathcal{D}$ is holomorphic in $\overline{W}$ with no zero on $\partial W$; then, so is $\mathcal{D} - \|\Gamma_n\|^2$ for $n$ large as $\|\Gamma_n\|^2 \to 0$. Let $\ell$ and $\ell_n$ be minimal degree polynomials, normalized by imposing $\|\ell\|_{\mathcal{T}} = \|\ell_n\|_{\mathcal{T}} = 1$, such that

$$ \frac{\mathcal{D}(\eta)}{\ell(p(\eta))} \quad \text{and} \quad \frac{\mathcal{D}(\eta) - \|\Gamma_n\|^2}{\ell_n(p(\eta))} $$

are holomorphic and non-vanishing in $W$. Since $\|\Gamma_n\|^2 \to 0$, the zeros of $\mathcal{D} - \|\Gamma_n\|^2$ in $W$ tend to those of $\mathcal{D}$ by Rouché's theorem, and so our normalization implies that $\ell_n \to \ell$ uniformly in $\overline{W}$ as $n \to \infty$. Hence, $(\mathcal{D} - \|\Gamma_n\|^2)/(\ell_n \circ p)$ converges to $\mathcal{D}/(\ell_n \circ p)$ uniformly in $\overline{W}$, in particular it is uniformly bounded away from zero there. Consequently, it follows from (3.23) that

$$ (3.24) \quad \log |(f - N(M_n) \circ p)(\eta)| \ge C + \log |\ell_n(p(\eta))|, \quad \eta \in \mathcal{T}', $$

for some finite constant $C$. Note that $\log |\ell_n| \le 0$ in $\mathbb{D}$ according to our normalization. Let us write $\ell_n(x) = a_n \prod_i (x - x_{i,n})$ and define the reciprocal polynomial $\tilde{\ell}_n$ of $\ell_n$ by

$$ \tilde{\ell}_n(x) := a_n \prod_i \begin{cases} x - x_{i,n} & \text{if } |x_{i,n}| \ge 1, \\ 1 - x \tilde{x}_{i,n} & \text{if } |x_{i,n}| < 1. \end{cases} $$

Clearly, $|\ell_n(\xi)| = |\tilde{\ell}_n(\xi)|$ for $|\xi| = 1$, and the maximum principle for harmonic functions implies that

$$ (3.25) \quad \int_{\mathcal{T}'} \log |\tilde{\ell}_n(p(\eta))| d\delta_z^{\mathcal{R}\setminus\Omega} \ge \log |\tilde{\ell}_n(p(z))|, \quad z \in \Omega, $$
since both sides of (3.25) are harmonic in $\Omega$ and have the trace $\log|\ell_n(p(\eta))|$ on $\mathcal{T}'$ while the left-hand side has zero trace on $\partial\Omega\setminus\mathcal{T}'$ and the right-hand side satisfies $\log|\ell_n \circ p| \le 0$ there. Thus, we get from (3.24) and (3.25) that

$$ (3.26) \qquad \begin{aligned} \int_{\mathcal{T}'} \frac{1}{n} \log|f - N(M_n) \circ p| d\delta_z^{\mathcal{R}\setminus\Omega} &\ge \frac{C}{n} + \frac{1}{n} \int_{\mathcal{T}'} \log|\tilde{\ell}_n(p(\eta))| d\delta_z^{\mathcal{R}\setminus\Omega} \\ &\ge \frac{C}{n} + \frac{1}{n} \log|\tilde{\ell}_n(p(z))|. \end{aligned} $$

As the last term goes to zero uniformly on $\Omega$ and since $\tilde{\ell}_n \to \tilde{\ell}$, where $\tilde{\ell}$ is the reciprocal polynomial of $\ell$ defined similarly to $\tilde{\ell}_n$, the estimate (3.20) now follows from (3.21), (3.22), and (3.26). $\square$

**3.6. Asymptotic Distributions of Poles and Zeros.** Recall the measures $\mu_n$ introduced in (3.10). Since these measures have mass at most 1, it follows from Helly's selection theorem [55, Theorem 0.1.3] that there exists a Borel measure $\mu'$, supported in $\mathbb{D}$ with mass at most 1, such that

$$ (3.27) \qquad \mu_n \xrightarrow{w^*} \mu' \quad \text{as} \quad \mathcal{N}' \ni n \to \infty, $$

perhaps at a cost of further restricting $\mathcal{N}'$, where $w^* \xrightarrow{\cdot} \mu$ stands for weak* convergence of finite (signed) measures (a sequence of Borel measures $\{\sigma_n\}$ on a locally compact space $X$ converges weak* to a measure $\sigma$ if $\int_X g d\sigma_n \to \int_X g d\sigma$ for every continuous function $g$ in $C_0(X)$, the completion of $C_c(X)$ in the supremum norm). Observe that $\mu'_{\mathbb{D}} = \mu$, where $\mu$ was defined as the vague limit $\mu_n$ in $\mathbb{D}$ along $\mathcal{N} \supseteq \mathcal{N}'$. In particular, $\mu_n \xrightarrow{w^*} \mu$ in $\mathbb{D}$.

**Lemma 3.3.** For any subsequence $\{n_m\}_{m=1}^{\infty} \subseteq \mathcal{N}'$, it holds that

$$ (3.28) \qquad \liminf_{m \to \infty} \frac{1}{n_m} \log |b_{n_m}^{\text{pole}}(z)|^{-1} \left\{ \begin{array}{l} \ge \\ = \end{array} \right\} g(\mu, \mathbb{D}; z), $$

where the inequality holds for every $z \in \mathbb{D}$ and equality holds for quasi every $z \in \mathbb{D}$.

*Proof.* Observe that $\frac{1}{n} \log |b_n^{\text{pole}}(z)|^{-1} = g(\mu_n, \mathbb{D}; z)$, see (3.9). Since $\mu_n \xrightarrow{w^*} \mu$ on $\mathbb{D}$, the conclusion follows from the *Principle of Descent* and the *Lower Envelope Theorem*, see Section A.7. $\square$

We cannot immediately get an analog of the previous lemma for the measures $\nu_n = \tilde{\nu}_n/n$, because we do not know if these Radon measures on $\mathbb{R}\setminus E_f$ have uniformly bounded masses. Instead, we shall study the asymptotic behavior of their Green potentials in the style of Lemmas 3.1 and 3.2.

**Lemma 3.4.** If the Radon measures $\sigma_n$ converge vaguely to $\sigma$, $K \subset E$ is compact, and $\sigma(\partial K) = 0$, then the restrictions $\sigma_{n|K}$ have uniformly bounded masses, $\sigma_{n|K} \xrightarrow{w^*} \sigma_{|K}$ on $K$, and $\sigma_n(\partial K) \to 0$.

*Proof.* For each $\varepsilon > 0$ there is an open set $V \supset \partial K$ such that $\sigma(V) < \varepsilon$ (by outer regularity of $\sigma$), and an open set $W$ with compact closure satisfying $V \supset \bar{W} \supset W \supset \partial K$ together with a continuous function $\varphi \ge 0$, supported in $V$, which is 1 on $\bar{W}$ (by Urysohn's lemma). Thus, $K_\varepsilon := K \setminus W$ is a compact subset of int $K$ (the interior of $K$) such that, for $n$ large enough that $|\int \varphi d\sigma_n - \int \varphi d\sigma| < \varepsilon$,

$$ \sigma_n(\partial K) \le \sigma_n(K \setminus K_\varepsilon) \le \sigma_n(W) \le \int \varphi d\sigma_n < \varepsilon + \int \varphi d\sigma \le 2\varepsilon. $$

Therefore $\int g d\sigma_n \to \int g d\sigma$ as $n \to \infty$ for any bounded continuous function $g$ on int $K$ [11, Proposition 6.18], and since $\sigma_n(\partial K) \to 0$ while $\sigma(\partial K) = 0$ it implies the weak* convergence of $\sigma_{n|K}$ to $\sigma_{|K}$. The uniform boundedness of the masses $\sigma_n(K) = \int_K 1 d\sigma_n$ now follows. $\square$

From now on we employ standard notation $\mathbb{D}_r := \{z : |z| < r\}$ and $\mathbb{T}_r := \partial \mathbb{D}_r$.
since both sides of (3.25) are harmonic in $\Omega$ and have the trace $\log|\ell_n(p(\eta))|$ on $\mathcal{T}'$ while the left-hand side has zero trace on $\partial\Omega\setminus\mathcal{T}'$ and the right-hand side satisfies $\log|\ell_n \circ p| \le 0$ there. Thus, we get from (3.24) and (3.25) that

$$ (3.26) \qquad \begin{aligned} \int_{\mathcal{T}'} \frac{1}{n} \log|f - N(M_n) \circ p| d\delta_z^{\mathcal{R}\setminus\Omega} &\ge \frac{C}{n} + \frac{1}{n} \int_{\mathcal{T}'} \log|\tilde{\ell}_n(p(\eta))| d\delta_z^{\mathcal{R}\setminus\Omega} \\ &\ge \frac{C}{n} + \frac{1}{n} \log|\tilde{\ell}_n(p(z))|. \end{aligned} $$

As the last term goes to zero uniformly on $\Omega$ and since $\tilde{\ell}_n \to \tilde{\ell}$, where $\tilde{\ell}$ is the reciprocal polynomial of $\ell$ defined similarly to $\tilde{\ell}_n$, the estimate (3.20) now follows from (3.21), (3.22), and (3.26). $\square$

**3.6. Asymptotic Distributions of Poles and Zeros.** Recall the measures $\mu_n$ introduced in (3.10). Since these measures have mass at most 1, it follows from Helly's selection theorem [55, Theorem 0.1.3] that there exists a Borel measure $\mu'$, supported in $\mathbb{D}$ with mass at most 1, such that

$$ (3.27) \qquad \mu_n \xrightarrow{w^*} \mu' \quad \text{as} \quad \mathcal{N}' \ni n \to \infty, $$

perhaps at a cost of further restricting $\mathcal{N}'$, where $w^* \xrightarrow{\cdot} \mu$ stands for weak* convergence of finite (signed) measures (a sequence of Borel measures $\{\sigma_n\}$ on a locally compact space $X$ converges weak* to a measure $\sigma$ if $\int_X g d\sigma_n \to \int_X g d\sigma$ for every continuous function $g$ in $C_0(X)$, the completion of $C_c(X)$ in the supremum norm). Observe that $\mu'_{|\mathbb{D}} = \mu$, where $\mu$ was defined as the vague limit $\mu_n$ in $\mathbb{D}$ along $\mathcal{N} \supseteq \mathcal{N}'$. In particular, $\mu_n \xrightarrow{w^*} \mu$ in $\mathbb{D}$.

**Lemma 3.3.** For any subsequence $\{n_m\}_{m=1}^\infty \subseteq \mathcal{N}'$, it holds that

$$ (3.28) \qquad \liminf_{m \to \infty} \frac{1}{n_m} \log |b_{n_m}^{\text{pole}}(z)|^{-1} \left\{ \begin{array}{l} \ge \\ = \end{array} \right\} g(\mu, \mathbb{D}; z), $$

where the inequality holds for every $z \in \mathbb{D}$ and equality holds for quasi every $z \in \mathbb{D}$.

*Proof.* Observe that $\frac{1}{n} \log |b_n^{\text{pole}}(z)|^{-1} = g(\mu_n, \mathbb{D}; z)$, see (3.9). Since $\mu_n \xrightarrow{w^*} \mu$ on $\mathbb{D}$, the conclusion follows from the *Principle of Descent* and the *Lower Envelope Theorem*, see Section A.7. $\square$

We cannot immediately get an analog of the previous lemma for the measures $\nu_n = \tilde{\nu}_n/n$, because we do not know if these Radon measures on $\mathcal{R}\setminus E_f$ have uniformly bounded masses. Instead, we shall study the asymptotic behavior of their Green potentials in the style of Lemmas 3.1 and 3.2.

**Lemma 3.4.** If the Radon measures $\sigma_n$ converge vaguely to $\sigma$, $K \subset E$ is compact, and $\sigma(\partial K) = 0$, then the restrictions $\sigma_{n|K}$ have uniformly bounded masses, $\sigma_{n|K} \xrightarrow{w^*} \sigma_{|K}$ on $K$, and $\sigma_n(\partial K) \to 0$.

*Proof.* For each $\varepsilon > 0$ there is an open set $V \supset \partial K$ such that $\sigma(V) < \varepsilon$ (by outer regularity of $\sigma$), and an open set $W$ with compact closure satisfying $V \supset \bar{W} \supset W \supset \partial K$ together with a continuous function $\varphi \ge 0$, supported in $V$, which is 1 on $\bar{W}$ (by Urysohn's lemma). Thus, $K_\varepsilon := K \setminus W$ is a compact subset of int $K$ (the interior of $K$) such that, for $n$ large enough that $|\int \varphi d\sigma_n - \int \varphi d\sigma| < \varepsilon$,

$$ \sigma_n(\partial K) \le \sigma_n(K \setminus K_\varepsilon) \le \sigma_n(W) \le \int \varphi d\sigma_n < \varepsilon + \int \varphi d\sigma \le 2\varepsilon. $$

Therefore $\int g d\sigma_n \to \int g d\sigma$ as $n \to \infty$ for any bounded continuous function $g$ on int $K$ [11, Proposition 6.18], and since $\sigma_n(\partial K) \to 0$ while $\sigma(\partial K) = 0$ it implies the weak* convergence of $\sigma_{n|K}$ to $\sigma_{|K}$. The uniform boundedness of the masses $\sigma_n(K) = \int_K 1 d\sigma_n$ now follows. $\square$

From now on we employ standard notation $\mathbb{D}_r := \{z : |z| < r\}$ and $\mathbb{T}_r := \partial \mathbb{D}_r$.
**Lemma 3.5.** There exist a subsequence $\mathcal{N}'' \subseteq \mathcal{N}'$ and a non-negative superharmonic function $u''(z)$ on $\mathcal{R}$ such that

$$ (3.29) \qquad \liminf_{m \to \infty} \frac{1}{n} \log |b_{n,m}^{\text{zero}}(z)|^{-1} \begin{cases} \geq \\ = \\ \geq \end{cases} u''(z) $$

for any $\{n_m\}_{m=1}^\infty \subseteq \mathcal{N}'$, where the inequality in (3.29) holds everywhere on $\mathcal{R}\setminus E_f$ while the equality needs only to hold quasi everywhere. When $u'' \neq +\infty$, it holds that

$$ (3.30) \qquad u'' = g(v'', \mathcal{R}; \cdot) + h'' $$

for some Radon measure $v''$ carried by $\mathcal{R}$ and some non-negative function $h''$ harmonic on $\mathcal{R}$.

*Proof.* Observe that $\frac{1}{n} \log |b_{n,m}^{\text{zero}}(z)|^{-1} = g(v_n, \Omega_m; z)$ according to (3.9). Below, we distinguish two cases: (i) when the measures $v_n$ possess a subsequence which is locally bounded in $\mathcal{R}\setminus E_f$, i.e., having uniformly bounded masses on each compact subset of $\mathcal{R}\setminus E_f$, and (ii) when there exists a compact set $K \subset \mathcal{R}\setminus E_f$ such that $v_n(K) \to \infty$ as $\mathcal{N}' \ni n \to \infty$.

In case (ii) relation (3.29) holds with $u'' \equiv \infty$ and $\mathcal{N}'' = \mathcal{N}'$ because $\min_{w \in K} g_{\Omega_\ell}(z, w) > 0$ for $z \in \Omega_\ell$ and every $\ell$ such that $K \subset \Omega_\ell$, and therefore

$$ g(v_{n'_m}, \Omega_m; z) \geq g(v_{n'_m}, \Omega_\ell; z) \geq v_{n'_m}(K) \min_{w \in K} g_{\Omega_\ell}(z, w) \to \infty $$

as $m \to \infty$ and $\{n'_m\}_{m=1}^\infty \subseteq \mathcal{N}'$, where the first inequality holds for $m \geq \ell$.

In case (i), the measures $v_n$ converge vaguely to $v^*$ in $\mathcal{R}\setminus E_f$ along $\mathcal{N}'$. Let $\{r_l\}_{l=1}^\infty$ be a positive real sequence increasing to 1 with $r_1$ large enough that $p(\partial\Omega_m \setminus \partial\mathcal{R}) \subset D_{r_1}$ and $v^*(p^{-1}(\mathbb{T}_{r_1})) = 0$ for each $l$. This is possible, because for $0 < a < b < 1$ the set $\cup_{a \leq r \leq b} p^{-1}(\mathbb{T}_r)$ is compact, so there are at most countably many $r \in [a, 1)$ with $v^*(p^{-1}(\mathbb{T}_r)) \neq 0$.

We now argue by double induction over $m$ and $l$: the reasoning below should be applied inductively in $m \geq 1$, so as to define a sequence of integers $\mathcal{N}_m$ for each $m$. Let $\Omega_{m,l} := \Omega_m \cap p^{-1}(D_{r_1})$ and proceed inductively in $l \geq 1$, starting with $\mathcal{N}_{m,0} := \mathcal{N}_{m-1}$ where $\mathcal{N}_0 = \mathcal{N}'$. Since $v_n(\bar{\Omega}_m) < \infty$ for each $n, m$ by definition of $v_n$, we can define

$$ (3.31) \qquad h_{n,m,l}(z) := g(v_n, \Omega_m; z) - g(v_{n|\bar{\Omega}_{m,l}}, \Omega_m; z), \quad z \in \Omega_{m,l}, $$

which is a non-negative harmonic function in $\Omega_{m,l}$. By Harnack's theorem, either there is a subsequence $\mathcal{N}_{m,l} \subseteq \mathcal{N}_{m,l-1}$ of indices $n$ along which $h_{n,m,l} \to h_{m,l}$, locally uniformly in $\Omega_{m,l}$, for some non-negative harmonic function $h_{m,l}$, or else $h_{n,m,l}$ tends to infinity with $n \in \mathcal{N}_{m,l-1}$, locally uniformly in $\Omega_{m,l}$. In the latter case, we set $\mathcal{N}_{m,l} := \mathcal{N}_{m,l-1}$ and $h_{m,l} \equiv +\infty$. Clearly, $h_{n,m,l} \geq h_{n,m,l+1}$ and so, for fixed $m$, either $h_{m,l} \equiv +\infty$ for all $l$ or the $h_{m,l}$ are finite for $l$ large enough. Let $\mathcal{N}_m$ be the diagonal of the table $\{\mathcal{N}_{m,l}\}_{l=1}^\infty$. Since $\mathcal{N}_m$ is eventually a subsequence of every $\mathcal{N}_{m,l}$, it holds that $h_{n,m,l} \to h_{m,l}$ as $\mathcal{N}_m \ni n \to \infty$ for every $l \geq 1$, locally uniformly in $\Omega_{m,l}$.

In another connection, since $v^*(\partial\Omega_{m,l}) = 0$ by construction and the $v_{n|\bar{\Omega}_{m,l}}$ have uniformly bounded mass over $n$ by the assumptions of the considered case, we deduce from Lemma 3.4 that

$$ v_{n|\bar{\Omega}_{m,l}} \xrightarrow{w^*} v^*_{|\bar{\Omega}_{m,l}} \quad \text{on } \bar{\Omega}_{m,l} \quad \text{as } \mathcal{N}_0 \ni n \to \infty. $$

Now, $v_{n|\bar{\Omega}_{m,l}}$ defines a measure on $\Omega_m$ in a natural way, and the weak* convergence above implies weak* convergence on $\Omega_m$, because every continuous function with compact support in $\Omega_m$ restricts to a continuous function on $\Omega_m \cap p^{-1}(D_{r_1})$ which itself extends to $\bar{\Omega}_{m,l} = \bar{\Omega}_m \cap p^{-1}(D_{r_1})$ continuously by zero. As $\Omega_m$ is a regular open set with compact closure on the surface $\mathcal{R}_*$, the Principle of Descent and the Lower Envelope Theorem yield for any subsequence $\mathcal{N}^* \subseteq \mathcal{N}_0$ that

$$ \liminf_{\mathcal{N}^* \ni n \to \infty} g(v_{n|\bar{\Omega}_{m,l}}, \Omega_m; z) \begin{cases} \geq \\ = \\ \geq \end{cases} g(v^*_{|\bar{\Omega}_{m,l}}, \Omega_m; z), $$

where the inequality holds everywhere in $\Omega_m$ and the equality may only hold quasi everywhere.
**Lemma 3.5.** There exist a subsequence $\mathcal{N}'' \subseteq \mathcal{N}'$ and a non-negative superharmonic function $u''(z)$ on $\mathcal{R}$ such that

$$ (3.29) \qquad \liminf_{m \to \infty} \frac{1}{n} \log |b_{n,m}^{\text{zero}}(z)|^{-1} \begin{cases} \geq \\ = \\ \geq \end{cases} u''(z) $$

for any $\{n_m\}_{m=1}^\infty \subseteq \mathcal{N}'$, where the inequality in (3.29) holds everywhere on $\mathcal{R}\setminus E_f$ while the equality needs only to hold quasi everywhere. When $u'' \neq +\infty$, it holds that

$$ (3.30) \qquad u'' = g(v'', \mathcal{R}; \cdot) + h'' $$

for some Radon measure $v''$ carried by $\mathcal{R}$ and some non-negative function $h''$ harmonic on $\mathcal{R}$.

*Proof.* Observe that $\frac{1}{n} \log |b_{n,m}^{\text{zero}}(z)|^{-1} = g(v_n, \Omega_m; z)$ according to (3.9). Below, we distinguish two cases: (i) when the measures $v_n$ possess a subsequence which is locally bounded in $\mathcal{R}\setminus E_f$, i.e., having uniformly bounded masses on each compact subset of $\mathcal{R}\setminus E_f$, and (ii) when there exists a compact set $K \subset \mathcal{R}\setminus E_f$ such that $v_n(K) \to \infty$ as $\mathcal{N}' \ni n \to \infty$.

In case (ii) relation (3.29) holds with $u'' \equiv \infty$ and $\mathcal{N}'' = \mathcal{N}'$ because $\min_{w \in K} g\Omega_\ell(z, w) > 0$ for $z \in \Omega_\ell$ and every $\ell$ such that $K \subset \Omega_\ell$, and therefore

$$ g(v_{n'_m}, \Omega_m; z) \geq g(v_{n'_m}, \Omega_\ell; z) \geq v_{n'_m}(K) \min_{w \in K} g\Omega_\ell(z, w) \to \infty $$

as $m \to \infty$ and $\{n'_m\}_{m=1}^\infty \subseteq \mathcal{N}'$, where the first inequality holds for $m \geq \ell$.

In case (i), the measures $v_n$ converge vaguely to $v^*$ in $\mathcal{R}\setminus E_f$ along $\mathcal{N}'$. Let $\{r_l\}_{l=1}^\infty$ be a positive real sequence increasing to 1 with $r_1$ large enough that $p(\partial\Omega_m \setminus \partial\mathcal{R}) \subset D_{r_1}$ and $v^*(p^{-1}(T_{r_1})) = 0$ for each $l$. This is possible, because for $0 < a < b < 1$ the set $\cup_{a \leq r \leq b} p^{-1}(T_r)$ is compact, so there are at most countably many $r \in [a, 1)$ with $v^*(p^{-1}(T_r)) \neq 0$.

We now argue by double induction over $m$ and $l$: the reasoning below should be applied inductively in $m \geq 1$, so as to define a sequence of integers $\mathcal{N}_m$ for each $m$. Let $\Omega_{m,l} := \Omega_m \cap p^{-1}(D_{r_l})$ and proceed inductively in $l \geq 1$, starting with $\mathcal{N}_{m,0} := \mathcal{N}_{m-1}$ where $\mathcal{N}_0 = \mathcal{N}'$. Since $v_n(\bar{\Omega}_m) < \infty$ for each $n, m$ by definition of $v_n$, we can define

$$ (3.31) \qquad h_{n,m,l}(z) := g(v_n, \Omega_m; z) - g(v_{n|\bar{\Omega}_{m,l}}, \Omega_m; z), \quad z \in \Omega_{m,l}, $$

which is a non-negative harmonic function in $\Omega_{m,l}$. By Harnack's theorem, either there is a subsequence $\mathcal{N}_{m,l} \subseteq \mathcal{N}_{m,l-1}$ of indices $n$ along which $h_{n,m,l} \to h_{m,l}$, locally uniformly in $\Omega_{m,l}$, for some non-negative harmonic function $h_{m,l}$, or else $h_{n,m,l}$ tends to infinity with $n \in \mathcal{N}_{m,l-1}$, locally uniformly in $\Omega_{m,l}$. In the latter case, we set $\mathcal{N}_{m,l} := \mathcal{N}_{m,l-1}$ and $h_{m,l} \equiv +\infty$. Clearly, $h_{n,m,l} \geq h_{n,m,l+1}$ and so, for fixed $m$, either $h_{m,l} \equiv +\infty$ for all $l$ or the $h_{m,l}$ are finite for $l$ large enough. Let $\mathcal{N}_m$ be the diagonal of the table $\{\mathcal{N}_{m,l}\}_{l=1}^\infty$. Since $\mathcal{N}_m$ is eventually a subsequence of every $\mathcal{N}_{m,l}$, it holds that $h_{n,m,l} \to h_{m,l}$ as $\mathcal{N}_m \ni n \to \infty$ for every $l \geq 1$, locally uniformly in $\Omega_{m,l}$.

In another connection, since $v^*(\partial\Omega_{m,l}) = 0$ by construction and the $v_{n|\bar{\Omega}_{m,l}}$ have uniformly bounded mass over $n$ by the assumptions of the considered case, we deduce from Lemma 3.4 that

$$ v_{n|\bar{\Omega}_{m,l}} \xrightarrow{w^*} v_{\lfloor \bar{\Omega}_{m,l} \rfloor}^* \quad \text{on } \bar{\Omega}_{m,l} \quad \text{as } \mathcal{N}_0 \ni n \to \infty. $$

Now, $v_{n|\bar{\Omega}_{m,l}}$ defines a measure on $\Omega_m$ in a natural way, and the weak* convergence above implies weak* convergence on $\Omega_m$, because every continuous function with compact support in $\Omega_m$ restricts to a continuous function on $\Omega_m \cap p^{-1}(D_{r_l})$ which itself extends to $\bar{\Omega}_{m,l} = \bar{\Omega}_m \cap p^{-1}(D_{r_l})$ continuously by zero. As $\Omega_m$ is a regular open set with compact closure on the surface $\mathcal{R}_*$, the Principle of Descent and the Lower Envelope Theorem yield for any subsequence $\mathcal{N}^* \subseteq \mathcal{N}_0$ that

$$ \liminf_{\mathcal{N}^* \ni n \to \infty} g(v_{n|\bar{\Omega}_{m,l}}, \Omega_m; z) \begin{cases} \geq \\ = \\ \geq \end{cases} g(v_{\lfloor \bar{\Omega}_{m,l} \rfloor}^*, \Omega_m; z), $$

where the inequality holds everywhere in $\Omega_m$ and the equality may only hold quasi everywhere.
In view of (3.31), the above inequality and the very definition of $N_m$ imply that

$$ (3.32) \quad \liminf_{N^* \ni n \to \infty} g(v_n, \Omega_m; z) \left\{ \begin{array}{l} \ge \varepsilon \\ = \\ \le \varepsilon \end{array} \right\} g(v_{\lfloor \Omega_{m,l} \rfloor}^*, \Omega_m; z) + h_{m,l}(z), \quad z \in \Omega_{m,l}, $$

along any subsequence $N^* \subseteq N_m$, where the inequality holds everywhere in $\Omega_{m,l}$ while the equality needs only to hold quasi everywhere. As the left-hand side of (3.32) does not depend on $l$ and the right-hand side is superharmonic, we get from the weak identity principle that successive right-hand sides are superharmonic continuations of each other when $l$ increases. Let $u_m(z)$ be the superharmonic function in $\Omega_m$ given on each $\Omega_{m,l}$ by the right-hand side of (3.32). Since a smooth function with compact support in $\Omega_m$ is eventually supported in $\Omega_{m,l}$ for large $l$, we get from the definition that either $u_m \equiv +\infty$ or $\Delta u_m = -v^*$. In the latter case, the Riesz representation theorem yields that

$$ (3.33) \quad u_m(z) = g(v^*, \Omega_m; z) + h_m(z), \quad z \in \Omega_m, $$

for some non-negative harmonic function $h_m$, which is the largest harmonic minorant of $u_m$.

Let $\tilde{N}$ be the diagonal of the table $\{N_m\}_{m=1}^\infty$. As $\tilde{N}$ is eventually a subsequence of each $N_m$, we get from (3.32) that for each $m$ and any subsequence $N^* \subseteq \tilde{N}$ it holds that

$$ (3.34) \quad \liminf_{N^* \ni n \to \infty} g(v_n, \Omega_m; z) \left\{ \begin{array}{l} \ge \varepsilon \\ = \\ \le \varepsilon \end{array} \right\} u_m(z), $$

where the inequality takes place everywhere in $\Omega_m$ and equality at least quasi everywhere. Because the left-hand side of (3.34) increases with $m$, we have that $u_m(z) \le u_{m+1}(z)$ for quasi every $z \in \Omega_m$. Thus, either $u_m \equiv +\infty$ for all $m$ large enough or else $u_m$ is finite quasi everywhere on $\Omega_m$ for all $m$. In the latter case, since $\Delta u_m = \Delta u_{m+1} \lfloor \Omega_m$ ($= -v_{\lfloor \Omega_m}^*$), we get that $u_{m+1} - u_m$ is harmonic on $\Omega_m$. Hence, $0 \le u_m \le u_{m+1}$ everywhere on $\Omega_m$, and so $u'' := \lim_m u_m$ is positive and superharmonic on $\mathbb{R} \setminus E_f$. If $u'' \equiv +\infty$ we are done, for we get (3.29) from (3.34) with $N'' = \tilde{N}$. Otherwise $u''$ is locally integrable and therefore (3.33), together with the Riesz representation theorem, imply that

$$ (3.35) \quad u''(z) = g(v^*, \mathbb{R} \setminus E_f; z) + \tilde{h}(z), \quad z \in \mathbb{R} \setminus E_f, $$

where $\tilde{h}(z)$ is a non-negative function that is the largest harmonic minorant of $u''$ on $\mathbb{R} \setminus E_f$.

As $E_f$ is polar and compact in $\mathbb{R}$, we deduce from the *Removability theorem* and the *Riesz representation theorem* that

$$ (3.36) \quad \tilde{h}(z) = h''(z) + g(\tilde{v}, \mathbb{R}; z), \quad z \in \mathbb{R} \setminus E_f, $$

where $h''$ is a non-negative harmonic function on $\mathbb{R}$ and $\tilde{v}$ a finite positive measure supported on $E_f$. Moreover, since for $z \in \mathbb{R} \setminus E_f$ the function $g_{\mathbb{R}}(z, \cdot) - g_{\mathbb{R} \setminus E_f}(z, \cdot)$ is non-negative harmonic on $\mathbb{R} \setminus E_f$ and bounded above near $E_f$, the *Removability theorem* for harmonic functions yields that $g_{\mathbb{R}}(z, \cdot) - g_{\mathbb{R} \setminus E_f}(z, \cdot) \equiv 0$ as it extends to a non-negative harmonic minorant of $g_{\mathbb{R}}(z, \cdot)$ on $\mathbb{R}$. Hence,

$$ (3.37) \quad g(v^*, \mathbb{R} \setminus E_f; z) = g(v^*, \mathbb{R}; z), \quad z \in \mathbb{R} \setminus E_f, $$

and equations (3.35)–(3.37) imply that $u''(z)$ extends superharmonically to the entire surface $\mathbb{R}$ by

$$ (3.38) \quad u''(z) = g(v'', \mathbb{R}; z) + h''(z), \quad v'' = v^* + \tilde{v}. $$

Now, for any subsequence $\{\tilde{n}_m\}_{m=1}^\infty \subseteq \tilde{N}$, it holds in view of (3.34) that for each $m_0 \in \mathbb{N}$ and $z \in \Omega_{m_0}$

$$ (3.39) \quad \liminf_{m \to \infty} g(v_{\tilde{n}_m}, \Omega_m; z) \ge \liminf_{m \to \infty} g(v_{\tilde{n}_m}, \Omega_{m_0}; z) \ge u_{m_0}(z), $$

and we obtain the inequality in (3.29) by letting $m_0$ tend to infinity.

Thus, it only remains to prove the equality quasi everywhere in (3.29) when $u''(z)$ is quasi everywhere finite. As before, the argument should be applied inductively on $m$ with $\tilde{N}_0 := \tilde{N}$. The functions $g_{n,l,m} := g(v_n, \Omega_l; \cdot) - g(v_n, \Omega_m; \cdot)$ are non-negative and harmonic in $\Omega_m$ for $l > m$. Therefore, by *Harnack's theorem*, there are subsequences $\tilde{N}_{m,l} \subseteq \tilde{N}_{m,l-1}$, $\tilde{N}_{m,0} := \tilde{N}_{m-1}$ such that
In view of (3.31), the above inequality and the very definition of $N_m$ imply that

$$ (3.32) \quad \liminf_{N^* \ni n \to \infty} g(v_n, \Omega_m; z) \left\{ \begin{array}{l} \ge \\ = \end{array} \right\} g(v^*_{\Omega_{m,l}}, \Omega_m; z) + h_{m,l}(z), \quad z \in \Omega_{m,l}, $$

along any subsequence $N^* \subseteq N_m$, where the inequality holds everywhere in $\Omega_{m,l}$ while the equality needs only to hold quasi everywhere. As the left-hand side of (3.32) does not depend on $l$ and the right-hand side is superharmonic, we get from the weak identity principle that successive right-hand sides are superharmonic continuations of each other when $l$ increases. Let $u_m(z)$ be the superharmonic function in $\Omega_m$ given on each $\Omega_{m,l}$ by the right-hand side of (3.32). Since a smooth function with compact support in $\Omega_m$ is eventually supported in $\Omega_{m,l}$ for large $l$, we get from the definition that either $u_m \equiv +\infty$ or $\Delta u_m = -v^*$. In the latter case, the Riesz representation theorem yields that

$$ (3.33) \qquad u_m(z) = g(v^*, \Omega_m; z) + h_m(z), \quad z \in \Omega_m, $$

for some non-negative harmonic function $h_m$, which is the largest harmonic minorant of $u_m$.

Let $\tilde{N}$ be the diagonal of the table $\{N_m\}_{m=1}^\infty$. As $\tilde{N}$ is eventually a subsequence of each $N_m$, we get from (3.32) that for each $m$ and any subsequence $N^* \subseteq \tilde{N}$ it holds that

$$ (3.34) \quad \liminf_{N^* \ni n \to \infty} g(v_n, \Omega_m; z) \left\{ \begin{array}{l} \ge \\ = \end{array} \right\} u_m(z), $$

where the inequality takes place everywhere in $\Omega_m$ and equality at least quasi everywhere. Because the left-hand side of (3.34) increases with $m$, we have that $u_m(z) \le u_{m+1}(z)$ for quasi every $z \in \Omega_m$. Thus, either $u_m \equiv +\infty$ for all $m$ large enough or else $u_m$ is finite quasi everywhere on $\Omega_m$ for all $m$. In the latter case, since $\Delta u_m = \Delta u_{m+1} \Omega_m (=-v^*_{\Omega_m})$, we get that $u_{m+1} - u_m$ is harmonic on $\Omega_m$. Hence, $0 \le u_m \le u_{m+1}$ everywhere on $\Omega_m$, and so $u'' := \lim_m u_m$ is positive and superharmonic on $\mathbb{R}\setminus E_f$. If $u'' \equiv +\infty$ we are done, for we get (3.29) from (3.34) with $N'' = \tilde{N}$. Otherwise $u''$ is locally integrable and therefore (3.33), together with the Riesz representation theorem, imply that

$$ (3.35) \qquad u''(z) = g(v^*, \mathbb{R}\setminus E_f; z) + \tilde{h}(z), \quad z \in \mathbb{R}\setminus E_f, $$

where $\tilde{h}(z)$ is a non-negative function that is the largest harmonic minorant of $u''$ on $\mathbb{R}\setminus E_f$.

As $E_f$ is polar and compact in $\mathbb{R}$, we deduce from the Removability theorem and the Riesz representation theorem that

$$ (3.36) \qquad \tilde{h}(z) = h''(z) + g(\tilde{v}, \mathbb{R}; z), \quad z \in \mathbb{R}\setminus E_f, $$

where $h''$ is a non-negative harmonic function on $\mathbb{R}$ and $\tilde{v}$ a finite positive measure supported on $E_f$. Moreover, since for $z \in \mathbb{R}\setminus E_f$ the function $g_{\mathbb{R}}(z, \cdot) - g_{\mathbb{R}\setminus E_f}(z, \cdot)$ is non-negative harmonic on $\mathbb{R}\setminus E_f$ and bounded above near $E_f$, the Removability theorem for harmonic functions yields that $g_{\mathbb{R}}(z, \cdot) - g_{\mathbb{R}\setminus E_f}(z, \cdot) \equiv 0$ as it extends to a non-negative harmonic minorant of $g_{\mathbb{R}}(z, \cdot)$ on $\mathbb{R}$. Hence,

$$ (3.37) \qquad g(v^*, \mathbb{R}\setminus E_f; z) = g(v^*, \mathbb{R}; z), \quad z \in \mathbb{R}\setminus E_f, $$

and equations (3.35)–(3.37) imply that $u''(z)$ extends superharmonically to the entire surface $\mathbb{R}$ by

$$ (3.38) \qquad u''(z) = g(v'', \mathbb{R}; z) + h''(z), \quad v'' = v^* + \tilde{v}. $$

Now, for any subsequence $\{\tilde{n}_m\}_{m=1}^\infty \subseteq \tilde{N}$, it holds in view of (3.34) that for each $m_0 \in \mathbb{N}$ and $z \in \Omega_{m_0}$

$$ (3.39) \quad \liminf_{m \to \infty} g(v_{\tilde{n}_m}, \Omega_m; z) \ge \liminf_{m \to \infty} g(v_{\tilde{n}_m}, \Omega_{m_0}; z) \ge u_{m_0}(z), $$

and we obtain the inequality in (3.29) by letting $m_0$ tend to infinity.

Thus, it only remains to prove the equality quasi everywhere in (3.29) when $u''(z)$ is quasi everywhere finite. As before, the argument should be applied inductively on $m$ with $\tilde{N}_0 := \tilde{N}$. The functions $g_{n,l,m} := g(v_n, \Omega_l; \cdot) - g(v_n, \Omega_m; \cdot)$ are non-negative and harmonic in $\Omega_m$ for $l > m$. Therefore, by Harnack's theorem, there are subsequences $\tilde{N}_{m,l} \subseteq \tilde{N}_{m,l-1}$, $\tilde{N}_{m,0} := \tilde{N}_{m-1}$ such that
$g_{n,l,m}$ 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

$$ f(z_1) = \lim_{\mathcal{N}_x \ni n \to \infty} N(M_n)(p(z_1)) = \lim_{\mathcal{N}_x \ni n \to \infty} N(M_n)(p(z_2)) = f(z_2) $$