<figure><img src="image_1.png" /></figure>

RESEARCH ARTICLE

Mathematika

# A decoupling interpretation of an old argument for Vinogradov's Mean Value Theorem

Brian Cook<fnref n="1" />

Kevin Hughes<fnref n="2" />,<fnref n="3" /> ID

Zane Kun Li<fnref n="4" /> ID

Akshat Mudgal<fnref n="5" /> ID

Olivier Robert<fnref n="6" />

Po-Lam Yung<fnref n="7" /> ID

**Correspondence**

Zane Kun Li, Department of
Mathematics, North Carolina State
University, Raleigh, NC 27695, USA.
Email: zkli@ncsu.edu

**Funding information**

EPSRC, Grant/Award Number:
EP/V521917/1; NSF, Grant/Award
Number: DMS-1902763; Ben Green's
Simons Investigator Grant, Grant/Award
Number: 376201; FWF-ANR,
Grant/Award Numbers: I 4945-N,
ANR-20-CE91-0006; Australian Research
Council, Grant/Award Number:
FT20010039; American Institute of
Mathematics

**Abstract**

We interpret into decoupling language a refinement of a 1973 argument due to Karatsuba on Vinogradov's mean value theorem. The main goal of our argument is to answer what precisely solution counting in older partial progress on Vinogradov's mean value theorem corresponds to in Fourier decoupling theory.

**MSC 2020**

42B15, 43A25, 43A70 (primary), 11L07 (secondary)

<footnote n="1">¹Department of Mathematics, Virginia Tech, Blacksburg, Virginia, USA</footnote>

<footnote n="2">²School of Mathematics, The University of Bristol, Bristol, UK</footnote>

<footnote n="3">³Heilbronn Institute for Mathematical Research, Bristol, UK</footnote>

<footnote n="4">⁴Department of Mathematics, North Carolina State University, Raleigh, North Carolina, USA</footnote>

<footnote n="5">⁵Mathematical Institute, University of Oxford, Oxford, UK</footnote>

<footnote n="6">⁶Université de Lyon, Université de Saint-Étienne, CNRS UMR 5208, Institut Camille Jordan, Saint-Étienne, France</footnote>

<footnote n="7">⁷Mathematical Sciences Institute, Australian National University, Canberra, ACT, Australia</footnote>
# 1 | INTRODUCTION

## 1.1 | Motivation

Let $s \ge 1$ and $k \ge 2$ be integers. For $X \ge 1$, let $J_{s,k}(X)$ be the number of solutions to the degree $k$ Vinogradov system in $2s$ variables:

$$x_1^j + x_2^j + \dots + x_s^j = y_1^j + y_2^j + \dots + y_s^j, \quad 1 \le j \le k, \qquad (1.1)$$

where all variables $x_1, \dots, x_s, y_1, \dots, y_s \in [1, X] \cap \mathbb{N}$. Nontrivial upper bounds for $J_{s,k}(X)$ were first studied by Vinogradov in 1935 [32] and such results are collectively referred to as Vinogradov's mean value theorem (VMVT) in the literature. The main conjecture in VMVT, now a theorem as of 2015, was that for every $\varepsilon > 0$ and $s, k \in \mathbb{N}$, one has

$$J_{s,k}(X) \lesssim_{s,k,\varepsilon} X^{\varepsilon} \left( X^s + X^{2s - \frac{k(k+1)}{2}} \right) \qquad (1.2)$$

for all $X \ge 1$. It is not hard to see that $J_{s,k}(X) \gtrsim_{s,k} X^s + X^{2s-k(k+1)/2}$ and applying Hölder's inequality, we may deduce (1.2) for all $s \in \mathbb{N}$ from the $s = k(k+1)/2$ case. VMVT plays an important role in understanding Waring's problem and the Riemann zeta function, see, for example, [11, 12, 19, 34]. When $k=2$, the main conjecture in VMVT is classical. In 2014, Wooley [35] proved the $k=3$ case of VMVT using the method of efficient congruencing (see also [20] for a shorter proof due to Heath-Brown). In 2015, the $k \ge 2$ case was proven by Bourgain, Demeter, and Guth in [3] using Fourier decoupling for the degree $k$ moment curve from which VMVT followed as a corollary. Finally, in 2017, Wooley [36], gave an alternative proof of (1.2) for all $k \ge 2$ using nested efficient congruencing.

After the proofs of VMVT using the Fourier method of decoupling [3] and the number theoretic method of efficient congruencing [36], it has been an interesting question to determine how these two methods are related and whether a “dictionary” between the two methods could be obtained. The study of this dictionary has led to new proofs of Fourier decoupling for the parabola [23], cubic moment curve [15], and the degree $k$ moment curve [16]; these having been inspired from the efficient congruencing arguments in [26, section 4], [20], and [36], respectively. Additionally, a decoupling interpretation of the study of VMVT over ellipspephic sets [1] led to a proof of Fourier decoupling for fractal sets on the parabola [5].

In this article, we revisit a particular classical VMVT which states that

$$J_{s,k}(X) \lesssim_{s,k} X^{2s - \frac{k(k+1)}{2} + \frac{1}{2}k^2 \left(1 - \frac{1}{k}\right)^{s/k}} \qquad (1.3)$$

for all $X \ge 1$ and $s = kl$ with $l \in \mathbb{N}$. This result should be compared to the supercritical $s \ge k(k+1)/2$ case in (1.2). For $s$ very large compared to $k$, we have an extra term $\frac{1}{2}k^2(1 - \frac{1}{k})^{s/k}$ in the exponent, which decays exponentially in $s$ for every fixed value of $k$, instead of an $\varepsilon$. The estimate (1.3) appears (for example) in Vaughan's book [31, chapter 5] and is a refinement of an argument of Karatsuba [22] from 1973 (see also Stechkin [27] from 1975). The loss of the $X^{\frac{1}{2}k^2(1 - \frac{1}{k})^{s/k}}$ comes from combining the subcritical estimate $J_{k,k}(X) \lesssim_k X^k$, which follows from the Newton-Girard identities, along with an iterative argument to derive estimates for $J_{s,k}(X)$ when $s$ is supercritical.
The main purpose of this paper is to illustrate how this refined argument of Karatsuba can be adapted to give a proof of a nonsharp Fourier decoupling inequality for the degree $k$ moment curve in the supercritical regime. The key difficulty that prevents the direct use of ideas from [15, 16, 23] is the heavy reliance on solution counting in (1.3). One of the main points of this article is to clarify the role of such solution counting arguments in the study of Fourier decoupling. The mechanism driving the solution counting arguments will allow us to prove the key Lemma 4.4, which concerns the geometry of Fourier supports of the functions appearing in our main Theorem 1.1.

As our goal is to clarify the role of solution counting in Fourier decoupling and Bourgain, Demeter, and Guth have already given the sharpest possible moment curve decoupling theorem in [3], we will work over $\mathbb{Q}_q$ rather than over $\mathbb{R}$. This will allow us to present the argument in the cleanest possible manner, free of technical difficulties arising from the inconvenience of the uncertainty principle in $\mathbb{R}^k$. See also [14] for another decoupling paper that works over $\mathbb{Q}_q$ rather than $\mathbb{R}$, there however, the authors use the observation that decoupling over $\mathbb{Q}_q$ is quantitatively more efficient than decoupling over $\mathbb{R}$ in terms of exponential sum estimates.

## Notation

As $k$ will be fixed, we will allow all constants to depend on $k$. Given two positive expressions $X$ and $Y$, we write $X \lesssim Y$ if $X \le CY$ for some constant $C$ that is allowed to depend on $k$. If $C$ depends on some additional parameter $A$, then we write $X \lesssim_A Y$. We write $X \sim Y$ if $X \le Y$ and $Y \le X$. By writing $f(x) = O(g(x))$, we mean $|f(x)| \lesssim g(x)$. We say that $f$ has Fourier support in a set $\Omega$ if its Fourier transform $\hat{f}$ is supported in $\Omega$.

To prepare the reader for the myriad of intervals that will occur later in Sections 4 and 5, there will be three types of interval lengths: intervals named with a “K” will be associated to the smallest scale $\delta$, intervals named with a “J” will be associated to the intermediate scale $\nu \approx \delta^{1/k}$, and intervals named with an “I” will be associated to the largest scale $\kappa \approx \delta^e$ (though on a first reading, it might be easier to set $\kappa = 1/q$). Finally, in the context of the decoupling constant $\mathfrak{D}_p(\delta)$, defined in (1.5), we call $p$ subcritical if $p < k(k+1)$ and $p$ supercritical if $p \ge k(k+1)$ (rather than the more accurate but slightly more clumsy “not subcritical”).

## 1.2 | Analysis over $\mathbb{Q}_q$ and decoupling

Fix a degree $k \ge 2$ and a prime number $q$ with $q > k$. We reserve the letter $p$ for the Lebesgue exponent in the main Theorem 1.1. We very briefly review the harmonic analysis over $\mathbb{Q}_q$ needed to set up the statement of decoupling. See also Section 2 and [14, section 2] for further discussion surrounding the harmonic analysis and basic geometric facts over $\mathbb{Q}_q$ that are useful in decoupling. Additionally see chapters 1 and 2 of [28] and chapter 1 (in particular sections 1 and 4) of [33] for a more complete discussion of analysis on $\mathbb{Q}_q$.

The field $\mathbb{Q}_q$ is the completion of $\mathbb{Q}$ under the $q$-adic norm, defined by $|0| = 0$ and $|q^a b/c| = q^{-a}$ if $a \in \mathbb{Z}$, $b, c \in \mathbb{Z} \setminus \{0\}$ and $q$ is relatively prime to both $b$ and $c$. Then $\mathbb{Q}_q$ can be identified (bijectively) with the set of all formal series

$$ \mathbb{Q}_q = \left\{ \sum_{j=k}^{\infty} a_j q^j : k \in \mathbb{Z}, a_j \in \{0, 1, \dots, q-1\} \text{ for every } j \ge k \right\}, $$
and the q-adic norm on $\mathbb{Q}_q$ satisfies $|\sum_{j=k}^{\infty} a_j q^j| = q^{-k}$ if $a_k \neq 0$. Strictly speaking we should be writing $|\cdot|_q$ instead of $|\cdot|$, but we omit this dependence as $q$ is fixed. The q-adic norm on $\mathbb{Q}_q$ induces a norm on $\mathbb{Q}_q^k$, which we denote also by $|\cdot|_q$ by abuse of notation, via $|(\xi_1, ..., \xi_k)| := \max_{1 \le i \le k} |\xi_i|$. Of particular importance is the ultrametric inequality: $|\xi + \eta| \le \max\{|\xi|, |\eta|\}$ with equality if $|\xi| \neq |\eta|$. An interval in $\mathbb{Q}_q$ is then a set of the form $\{\xi \in \mathbb{Q}_q : |\xi - a| \le r\}$, where $a \in \mathbb{Q}_q$ and $r \ge 0$; $r$ will then be called the length of the interval. We also will use $|I|$ to denote the length of an interval $I$. The ring of integers $\mathbb{Z}_q$ coincides with the unit interval $\{\xi \in \mathbb{Q}_q : |\xi| \le 1\}$. A cube in $\mathbb{Q}_q^k$ of side length $r$ is then a product of $k$ intervals in $\mathbb{Q}_q$ of lengths $r$. We will work with Schwartz functions defined on $\mathbb{Q}_q^k$ (i.e., finite linear combinations of characteristic functions of cubes in $\mathbb{Q}_q^k$). The Fourier transform of such a function $f$ will be given by

$$ \hat{f}(\xi) := \int_{\mathbb{Q}_q^k} f(x)\chi(-x \cdot \xi)dx, $$

where $\chi$ is a fixed element in the Pontryagin dual $\widehat{\mathbb{Q}}_q$ of $\mathbb{Q}_q$ that restricts to the principal character on the additive subgroup $\mathbb{Z}_q$ and restricts to a nonprincipal character on the additive subgroup $q^{-1}\mathbb{Z}_q$, $x \cdot \xi = \sum_{i=1}^k x_i \xi_i$ if $x = (x_1, ..., x_k)$ and $\xi = (\xi_1, ..., \xi_k)$, and $dx$ is the Haar measure on the additive group $\mathbb{Q}_q^k$ normalized so that $\int_{\mathbb{Z}_q^k} dx = 1$. One key property of the Fourier transform that we will use is that $1_{\widehat{\mathbb{Z}}_q} = 1_{\mathbb{Z}_q}$, that is, the Fourier transform of the unit ball is the unit ball, see [33, p. 42] for a proof.

We are interested in the unit moment curve

$$ \gamma(t) := (t, t^2, ..., t^k), \quad |t| \le 1. $$

For $\delta \in q^{-N}$ and any interval $I \subset \mathbb{Q}_q$ with length $\ge \delta$, let $P_\delta(I)$ be a partition of $I$ into intervals of length $\delta$. Write $P_\delta$ for $P_\delta(\mathbb{Z}_q)$. To each interval $I \subset \mathbb{Z}_q$, one associates a parallelepiped

$$ \theta_I := \{ \gamma(a) + \sum_{j=1}^{k} t_j \gamma^{(j)}(a) \in \mathbb{Q}_q^k : |t_j| \le |I|^j \text{ for all } 1 \le j \le k \} $$

of dimensions $|I| \times |I|^2 \times \dots \times |I|^k$, where $a \in I$; this parallelepiped is independent of the choice of $a \in I$. Note that $\bigcup_{K \in P_\delta} \theta_K$ is a covering of a $\delta^k$ neighborhood of the unit moment curve (in fact it covers a suitable anisotropic neighborhood of that curve). One also associates to each $K \in P_\delta$ a cube

$$ \tau_K := \{(\xi_1, ..., \xi_k) \in \mathbb{Q}_q^k : |\xi_j - a^j| \le \delta \text{ for all } 1 \le j \le k\} \quad (1.4) $$

of side length $\delta$, where $a \in K$; again this is independent of the choice of $a \in K$. Note that for each $K \subset P_\delta$, the ultrametric inequality gives that $\theta_K \subset \tau_K$.

For an interval $I \subset \mathbb{Z}_q$, let $f_I$ be defined such that $\hat{f}_I := \hat{f} \cdot 1_{I \times \mathbb{Q}_q^{k-1}}$. For $p \ge 2$ and $\delta \in q^{-N}$, let $\mathfrak{D}_p(\delta)$ be the smallest constant such that the inequality

$$ \|f\|_{L^p(\mathbb{Q}_q^k)} \le \mathfrak{D}_p(\delta) \left( \sum_{K \in P_\delta} \|f_K\|_{L^p(\mathbb{Q}_q^k)}^2 \right)^{1/2} \quad (1.5) $$
holds for every Schwartz function $f$ on $\mathbb{Q}_q^k$ with its Fourier transform $\hat{f}$ supported on $\bigcup_{K \in P_\delta} \theta_K$. Note that $f = \sum_{K \in P_\delta} f_K$. Bourgain, Demeter, and Guth [3] showed that

$$ \mathcal{D}_p(\delta) \lesssim_{\varepsilon, p, q} \delta^{-\varepsilon} \left( 1 + \delta^{-\left(\frac{1}{2} - \frac{k(k+1)}{2p}\right)} \right), \quad (1.6) $$

and this estimate is sharp. Strictly speaking [3] proves a decoupling theorem over $\mathbb{R}$ rather than over $\mathbb{Q}_q$, but the same proof can be used to derive (1.6). Choosing $f$ to be a sum of Dirac deltas immediately implies (1.2).

## 1.3 | The main result

By interpreting the refinement of Karatsuba's argument for (1.3) into decoupling language, our main result is then the following Fourier decoupling analogue of (1.3). In the same way that (1.3) is a weaker partial result toward (1.2), Theorem 1.1 and Corollary 1.2 should be viewed as the analogous weaker counterpart of the sharp bound (1.6).

**Theorem 1.1.** Let $p_0 \in 2\mathbb{N}$ be an even integer and let $c(p_0) \ge 0$ be such that

$$ \mathcal{D}_{p_0}(\delta) \le C_1 \delta^{-\left(\frac{1}{2} - \frac{k(k+1)}{2p_0}\right) - \frac{c(p_0)}{p_0} \left(1 - \frac{1}{k}\right)^{p_0/(2k)}} \quad \text{for all } \delta \in q^{-\mathbb{N}} \quad (1.7) $$

where $C_1$ is independent of $\delta$. If $p \in p_0 + 2k\mathbb{N}$ and $0 < \varepsilon < 1$, then

$$ \mathcal{D}_p(\delta) \lesssim_{p, \varepsilon, C_1} q^{a(p, p_0)/p} \delta^{-\left(\frac{1}{2} - \frac{k(k+1)}{2p}\right) - \frac{c(p_0)}{p} \left(1 - \frac{1}{k}\right)^{p/(2k)} - \varepsilon} \quad \text{for all } \delta \in q^{-\mathbb{N}}, \quad (1.8) $$

where

$$ a(p, p_0) := \left(\frac{p-p_0}{2k}\right) \left(\frac{p_0}{2} + \frac{k^2+7k-4}{2}\right) + \frac{k}{2} \left(\frac{p-p_0}{2k}\right) \left(\frac{p-p_0}{2k} + 1\right). \quad (1.9) $$

As $\mathcal{D}_p(\delta) \ge 1$ for all $p$, (1.7) implies that $c(p_0)$, $k$, and $p_0$ are such that

$$ \frac{1}{2} - \frac{k(k+1)}{2p_0} + \frac{c(p_0)}{p_0} \left(1 - \frac{1}{k}\right)^{\frac{p_0}{2k}} \ge 0. \quad (1.10) $$

It is also known that $\mathcal{D}_{2k}(\delta) \lesssim_{\varepsilon} \delta^{-\varepsilon}$ for any $\varepsilon > 0$, see, for example, [8, Exercise 11.19] for the Euclidean case; we provide a proof for the case over $\mathbb{Q}_q$ in the Appendix for the convenience of the reader. We also remark that [21] proved, in the case of local fields, a related square function estimate with a bound independent of $\delta$ if the $f_K$'s are Fourier supported in a $\delta^k$ neighborhood of $\gamma(K)$; see also [13] and [2] for similar estimates. Choosing $p_0 = 2k$ and $c(p_0) = k^2/2 + \varepsilon$ for any $\varepsilon > 0$ in applying Theorem 1.1 we obtain:

**Corollary 1.2.** Let $p \in 2k\mathbb{N}$ and $0 < \varepsilon < 1$. Then

$$ \mathcal{D}_p(\delta) \lesssim_{p, \varepsilon} q^{O(k+p/k)} \delta^{-\left(\frac{1}{2} - \frac{k(k+1)}{2p}\right) - \frac{k^2}{2p} \left(1 - \frac{1}{k}\right)^{p/(2k)} - \varepsilon} \quad \text{for all } \delta \in q^{-\mathbb{N}}, $$

where the implied constant in the exponent of $q$ is absolute (and independent of $k$).
The exponent of $q$ in Corollary 1.2 is more precisely $\frac{a(p,2k)}{p} = (\frac{1}{2k} - \frac{1}{p})^{\frac{k^2+9k-4}{2}} + \frac{1}{4}(\frac{p}{2k} - 1)$, but we opt to write it as above because it more clearly illustrates what the main terms are. Note that the hypothesis in Theorem 1.1 is always satisfied if $p_0$ is any fixed exponent $\ge 2$ and $c(p_0)$ is chosen large enough. One can view Theorem 1.1 as a way of upgrading trivial $l^2L^{p_0}$ decoupling at say some subcritical $p$ to $l^2L^p$ decoupling for all large $p$ with only a loss that decreases exponentially as $p \to +\infty$. Of course, if one already knew the sharp estimate in the critical $p_0 = k(k+1)$ case, then Theorem 1.1 implies that we know the sharp decoupling estimate for all $p \in k(k+1) + 2k\mathbb{N}$. However, this already follows from interpolating the critical estimate with the trivial $l^2L^\infty$ decoupling estimate.

Though Corollary 1.2 implies (1.3) with an extra $X^\epsilon$ that comes from the $\delta^{-\epsilon}$ factor in Corollary 1.2, Corollary 1.2 is more general and this extra $\delta^{-\epsilon}$ term comes from needing some additional uniformity in the case of the general $f$ Fourier supported in $\bigcup_{K \in P_\delta} \theta_K$ and an application of the broad-narrow argument to get around the use of the Prime Number Theorem in the proof of (1.3) (see Subsection 4.1.1). See Subsections 3.5 and 5.1 for some more discussion comparing the VMVT case and the general $f$ decoupling case.

We end with some discussion about how the proof of Corollary 1.2 (and Theorem 1.1) contrasts with modern decoupling proofs of degree $k$ moment curve decoupling [3, 16] that prove (1.6). Unlike the argument in [3, 16], we are missing any lower dimensional decoupling input and while we do use induction on scales, the iteration itself is unique in that it iterates on the $p$ in $l^2L^p$ decoupling. Schematically, the iteration to prove Theorem 1.1 controls $l^2L^p$ decoupling by $l^2L^{p-2k}$ decoupling at a larger scale. After $O(p/k)$ steps, we are reduced to $l^2L^{2k}$ decoupling for the degree $k$ moment curve which follows (essentially) from the Newton-Girard identities. The iteration is surprisingly efficient when it controls $l^2L^p$ decoupling by $l^2L^{p-2k}$ decoupling as long as both $p$ and $p-2k$ are supercritical. However, after about $\frac{1}{2k}(p - \frac{k(k+1)}{2})$ steps, we enter the subcritical regime for which the iteration becomes inefficient and this is why we accrue an additional $\delta^{-\frac{k^2}{2p}(1-\frac{1}{k})^{p/(2k)}}$ term. When $k=2$, the argument for Corollary 1.2 uses $O(p)$ steps to prove a weak nonsharp $l^2L^p$ decoupling estimate. This is to be compared to the modern proof of decoupling for the parabola where to prove the sharp critical $l^2L^6$ decoupling, one uses $O(\epsilon^{-1})$ many steps (see, for example, the proof of [23, Lemma 2.12]). In the harmonic analysis literature, iterating on $p$ is not a new idea as such an argument was already used by Drury [9] to prove cubic moment curve restriction, though we believe this is the first time such an argument has appeared in the decoupling literature. See also [25] by the fourth author for a similar idea in the additive combinatorics literature that was recently used to obtain diameter free estimates for the quadratic VMVT.

Additionally, at each iterative step, three scales are key: the smallest scale $\delta$, the intermediate scale $\delta^{1/k}$, and the largest scale 1 (though strictly speaking in our proof the largest scale is actually $\delta^\epsilon$ rather than 1 for technical reasons). This can be compared to [3, 16] which uses scales $\delta$, $\delta^\epsilon$ and 1.

This paper is organized as follows: In Section 2, we review some basic geometric and harmonic analysis facts in $\mathbb{Q}_q$ that will be used throughout this paper. In Section 3, we review the refinement of the 1973 argument of Karatsuba at a high level. In Section 4, we prove Lemma 4.2 which is the main lemma that is used to prove Theorem 1.1. This is accomplished via combining a standard broad-narrow argument in Subsection 4.1.1 and some geometric properties of the moment curve that use the Newton-Girard identites, see Lemma 4.4. In Section 5, we dyadically pigeonhole to obtain some uniformity in our estimates and prove Theorem 1.1 and Corollary 1.2. Finally, in the Appendix, we include a proof of $\mathfrak{D}_{2k}(\delta) \lesssim \delta^{-\epsilon}$ for completeness.
## 2 | WAVEPACKET DECOMPOSITION AND SOME BASIC GEOMETRIC FACTS

Throughout this paper, we will make use of wavepacket decomposition which allows us to decompose a function $f$, which is Fourier supported in some $\theta_K$, into linear combinations of indicator functions of translates of the parallelepiped “dual” to $\theta_K$. That the $q$-adic character $\chi$ is trivial on $\mathbb{Z}_q$ gives a much cleaner wavepacket decomposition when working over $\mathbb{Q}_q$ than over $\mathbb{R}$. See [30, section 3] or [17, section 2.4] for some discussion about wavepacket decomposition over $\mathbb{R}$ in the context of the paraboloid (though the same ideas apply for the degree $k$ moment curve).

Fix $\delta \in q^{-N}$. It will be convenient to introduce the shorthand

$$ \theta_{\delta} := \delta \mathbb{Z}_{q} \times \delta^{2} \mathbb{Z}_{q} \times \cdots \times \delta^{k} \mathbb{Z}_{q} $$

and

$$ T_{\delta} := \delta^{-1} \mathbb{Z}_{q} \times \delta^{-2} \mathbb{Z}_{q} \times \cdots \times \delta^{-k} \mathbb{Z}_{q}. $$

They are dual to each other in the sense that

$$ T_{\delta} = \{x \in \mathbb{Q}_{q}^{k} : |x \cdot \xi| \le 1 \text{ for all } \xi \in \theta_{\delta}\}. $$

Since for any $1 \le j \le k$, any interval in $\mathbb{Q}_q$ of length $\delta^j$ is the disjoint union of $\delta^{-(k-j)}$ many intervals of length $\delta^k$, it follows that $\theta_\delta$ is the disjoint union of $\delta^{-\frac{k(k-1)}{2}}$ many cubes of side lengths $\delta^k$ in $\mathbb{Q}_q^k$. Similarly, any cube in $\mathbb{Q}_q^k$ of side length $\delta^{-k}$ is a disjoint union of $\delta^{-\frac{k(k-1)}{2}}$ many translates of $T_\delta$.

Now for $a \in \mathbb{Z}_q$, let $M_a$ be the $k \times k$ lower-triangular matrix given by

$$ M_a = (\gamma'(a) \ \gamma''(a) \ \cdots \ \gamma^{(k)}(a)) $$

where we view $\gamma^{(j)}(a)$ as a column vector. Then for any $K \in P_\delta$, we have

$$ \theta_K = \gamma(a) + M_a \theta_\delta \quad (2.1) $$

for any $a \in K$. In fact, the right-hand side is independent of $a \in K$ because if $b \in K$, then

$$ \gamma(b) = \gamma(a) + \sum_{j=1}^{k} (j!)^{-1} \gamma^{(j)}(a) (b-a)^j \in \gamma(a) + M_a \theta_\delta, $$

and

$$ M_a = M_b \begin{pmatrix} 1 & 0 & \cdots & 0 \\ (1!)^{-1}(b-a) & 1 & \cdots & 0 \\ (2!)^{-1}(b-a)^2 & (1!)^{-1}(b-a) & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ ((k-1)!)^{-1}(b-a)^{k-1} & ((k-2)!)^{-1}(b-a)^{k-2} & \cdots & 1 \end{pmatrix}, \quad (2.2) $$
where the second matrix on the right-hand side preserves $\theta_{\delta} = \delta\mathbb{Z}_q \times \delta^2\mathbb{Z}_q \times \cdots \times \delta^k\mathbb{Z}_q$ (here we have used the fact that $|k!| = 1$ in $\mathbb{Q}_q$ because $q > k$).

For $K \in P_{\delta}$ and any $a \in K$, let $T_{0,K}$ be the dual parallelepiped to $\theta_K$ centered at the origin given by

$$T_{0,K} = \{x \in \mathbb{Q}_q^k : |x \cdot (\xi - \gamma(a))| \le 1 \text{ for all } \xi \in \Theta_K\}.$$

Using (2.1), it is not hard to see that

$$T_{0,K} = \{x \in \mathbb{Q}_q^k : |x \cdot \gamma^{(j)}(a)| \le \delta^{-j} \text{ for all } 1 \le j \le k\} \\ = \{x \in \mathbb{Q}_q^k : M_a^T x \in T_{\delta}\} = M_a^{-T} T_{\delta}$$

for any $a \in K$. This parallelepiped depends only on $K$ but not on the choice of $a \in K$, as (2.2) shows that

$$M_a^{-T} = M_b^{-T} \begin{pmatrix} 1 & O(\delta) & O(\delta^2) & \dots & O(\delta^{k-1}) \\ 0 & 1 & O(\delta) & \dots & O(\delta^{k-2}) \\ 0 & 0 & 1 & \dots & O(\delta^{k-3}) \\ \vdots & & \vdots & & \vdots \\ 0 & 0 & 0 & \dots & 1 \end{pmatrix},$$

where $O(\delta^j)$ is some number in $\mathbb{Q}_q$ with norm $\le \delta^j$, and the second matrix on the right-hand side is a bijection that preserves $T_{\delta}$ by the ultrametric inequality.

**Lemma 2.1.** Let $\delta \in q^{-\mathbb{N}}$ and fix $K \in P_{\delta}$. Then

(i) $\theta_K - \theta_K$ is the disjoint union of $\delta^{-\frac{k(k-1)}{2}}$ cubes of side lengths $\delta^k$, and

(ii) every cube of side length $\delta^{-k}$ in $\mathbb{Q}_q^k$ is the disjoint union of $\delta^{-\frac{k(k-1)}{2}}$ many translates of $T_{0,K}$.

*Proof.*

(i) Recall that $\theta_{\delta}$ is the disjoint union of $\delta^{-\frac{k(k-1)}{2}}$ cubes of side lengths $\delta^k$. As $M_a$ is a bijection that maps cubes of side length $\delta^k$ to cubes of side length $\delta^k$ for any $a \in K$, and $\theta_K - \theta_K = M_a\theta_{\delta}$ for any $a \in K$, the assertion follows. Note that $\theta_K - \theta_K$ is just a translation of $\theta_K$ to the origin.

(ii) Recall that any cube in $\mathbb{Q}_q^k$ of side length $\delta^{-k}$ is a disjoint union of $\delta^{-\frac{k(k-1)}{2}}$ many translates of $T_{\delta}$. As $M_a^{-T}$ is a bijection that maps cubes of side length $\delta^{-k}$ to cubes of side length $\delta^{-k}$ for any $a \in K$, and $T_{0,K} = M_a^{-T}T_{\delta}$ for any $a \in K$, the assertion follows. $\square$

From Lemma 2.1(ii), we may deduce that translates of $T_{0,K}$ tile $\mathbb{Q}_q^k$, we denote the collection of such translates by $\mathbb{T}(K)$. We are now ready to state the version of wavepacket decomposition that we will use.

**Lemma 2.2 (Wavepacket decomposition).** Let $\delta \in q^{-\mathbb{N}}$ and fix $K \in P_{\delta}$. Let $g$ be a Schwartz function with Fourier transform supported in $\theta_K$. Then $|g|$ is constant on every $T \in \mathbb{T}(K)$, and $\widehat{g1_T}$ is supported
on $\theta_K$ for every $T \in \mathbb{T}(K)$. Hence, it is natural to write

$$g = \sum_{T \in \mathbb{T}(K)} g1_T, \qquad (2.3)$$

where each term $g1_T$ (which we will call a “wavepacket”) is Fourier supported on $\theta_K$ and has constant modulus on every $T \in \mathbb{T}(K)$. It also follows that if $\mathcal{T}$ is any subset of $\mathbb{T}(K)$, then $\sum_{T \in \mathcal{T}} g1_T$ is Fourier supported in $\theta_K$.

*Proof.* First, to prove that $|g|$ is constant on any translates of $T_{0,K}$, one only needs to prove the case when $\delta = 1$, $K = \mathbb{Z}_q$, and then apply a change of variables, but we opt for a more explicit proof. We will show that $|g(x)|$ is constant for all $x \in A + T_{0,K}$ for any $A \in \mathbb{Q}_q^k$. By Fourier inversion we have that

$$ \begin{aligned} |g(x)| &= |\int_{\theta_K} \hat{g}(\xi) \chi(\xi \cdot x) d\xi| \\ &= |\int_{|t_1| \le \delta, \dots, |t_k| \le \delta^k} \hat{g}\left(\gamma(a) + \sum_{j=1}^k t_j \gamma^{(j)}(a)\right) \chi\left(\left[\gamma(a) + \sum_{j=1}^k t_j \gamma^{(j)}(a)\right] \cdot x\right) dt| \\ &= |\int_{|t_1| \le \delta, \dots, |t_k| \le \delta^k} \hat{g}(\gamma(a) + M_a t) \chi(M_a^T x \cdot t) dt|. \end{aligned} $$

For $x \in A + T_{0,K}$, we write $x = A + M_a^{-T}y'$ where $|y'_j| \le \delta^{-j}$ for $j = 1, 2, \dots, k$. Therefore,

$$ \begin{aligned} |g(x)| &= |\int_{|t_1| \le \delta, \dots, |t_k| \le \delta^k} \hat{g}(\gamma(a) + M_a t) \chi(M_a^T A \cdot t) \chi(y' \cdot t) dt| \\ &= |\int_{|t_1| \le \delta, \dots, |t_k| \le \delta^k} \hat{g}(\gamma(a) + M_a t) \chi(M_a^T A \cdot t) dt|, \end{aligned} $$

where we have used that $y' \cdot t \in \mathbb{Z}_q$, and so $\chi(y' \cdot t) = 1$. The right-hand side is then independent of $y'$ and so the above equality is true for all $x \in A + T_{0,K}$. In particular, this shows that $|g|$ is a constant on $A + T_{0,K}$. This constant depends on $K$, $g$ and $A$, but is a constant nonetheless.

Next, to prove that $\widehat{g1_T}$ is supported on $\theta_K$, it suffices to observe that $\widehat{g1_T} = \hat{g} * \widehat{1_T}$, and that $\widehat{1_T}$ is supported on $\theta_K - \theta_K$ for every $T \in \mathbb{T}(K)$: in fact, for every $T \in \mathbb{T}(K)$, $\widehat{1_T}$ is a modulation of $\widehat{1}_{T_{0,K}}$, and if $a$ is any point in $K$, then $T_{0,K} = M_a^{-T}T_\delta$. It follows that

$$ \begin{aligned} \widehat{1}_{T_{0,K}}(\xi) &= \int_{M_a^{-T}T_\delta} \chi(-x \cdot \xi)dx \\ &= \det(M_a)^{-1} \int_{T_\delta} \chi(-M_a^{-T}y \cdot \xi)dy = \det(M_a)^{-1} \delta^{-k(k+1)/2} \widehat{1}_{\theta_\delta}(M_a^{-1}\xi) \end{aligned} $$

is supported on $M_a\theta_\delta = \theta_K - \theta_K$. Finally, the decomposition (2.3) follows because parallelepipeds in $\mathbb{T}(K)$ tile $\mathbb{Q}_q^k$. This completes the proof of the lemma. $\square$
# 3 | SKETCH OF THE KARATSUBA ARGUMENT

Before we dive into the proof of Theorem 1.1, we review the proof of (1.3) with an eye toward interpreting each step into decoupling language. See also, for example, [31, section 5.1] or [29, Theorem 13 - Lemma 21] for more details of the number theoretic argument. Just for this section, we revert back to calling *p* a prime so as to best match these references.

## 3.1 | Step 1: Introducing some *p*-adic separation

Given $X \ge 1$, one finds, using the Prime Number Theorem, a prime $p \sim X^{1/k}$ such that $J_{s,k}(X)$ is controlled by $J_{s,k}(X, p)$, where $J_{s,k}(X, p)$ is defined to be the number of solutions $(x_1, ..., x_s, y_1, ..., y_s) \in ([1, X] \cap \mathbb{N})^{2s}$ to (1.1) with the additional condition that $x_1, ..., x_k$ are pairwise distinct mod $p$. As $p$ is rather large, this is a rather mild condition and so we heuristically should still expect $J_{s,k}(X) \approx J_{s,k}(X, p)$. The benefit of this extra *p*-adic separation (transversality) in these $2k$ variables is that we will get to apply Linnik's lemma (in Step 3, (3.3)) which will up to permutation uniquely determine these variables.

## 3.2 | Step 2: Applying the union bound/Hölder

We now write $J_{s,k}(X, p)$ as

$$ \int_{[0,1]^{2s}} \left| \sum_{\substack{a_1, ..., a_k \pmod p \\ a_i \text{ pairwise distinct}}} \prod_{j=1}^k \sum_{\substack{n_j \equiv a_j \pmod p \\ 1 \le n_j \le X}} e(n_j \alpha_1 + \dots + n_j^k \alpha_k) \right|^2 \left| \sum_{1 \le n \le X} e(n \alpha_1 + \dots + n^k \alpha_k) \right|^{2s-2k} d\alpha. $$

Write $|\sum_{1 \le n \le X} |^{2s-2k} = |\sum_a \pmod p \sum_{n \equiv a \pmod p} |^{2s-2k}$ and apply Hölder's inequality to control the above by

$$ p^{2s-2k} \max_a \pmod p \int_{[0,1]^{2s}} \left| \sum_{\substack{a_1, ..., a_k \pmod p \\ a_i \text{ pairwise distinct}}} \prod_{j=1}^k \sum_{\substack{n_j \equiv a_j \pmod p \\ 1 \le n_j \le X}} e(n_j \alpha_1 + \dots + n_j^k \alpha_k) \right|^2 \times \\ \left| \sum_{\substack{n \equiv a \pmod p \\ 1 \le n \le X}} e(n \alpha_1 + \dots + n^k \alpha_k) \right|^{2s-2k} d\alpha. \tag{3.1} $$

Denote the integral above to be $J_{s,k}(X, p, a)$. This expression counts the number of solutions $(x_1, ..., x_s, y_1, ..., y_s) \in ([1, X] \cap \mathbb{N})^{2s}$ to (1.1) with $x_1, ..., x_k$ pairwise distinct mod $p$, $y_1, ..., y_k$ pairwise distinct mod $p$, and $x_{k+1} \equiv \dots \equiv x_s \equiv y_{k+1} \equiv \dots \equiv y_s \equiv a \pmod p$.

## 3.3 | Step 3: Solution counting

Translation invariance of the Vinogradov system implies that we may bound $J_{s,k}(X, p, a)$ by $J_{s,k}(X, p, 0)$. Rearrange the Vinogradov system (1.1) as

$$ x_{k+1}^j + \dots + x_s^j - y_{k+1}^j - \dots - y_s^j = y_1^j + \dots + y_k^j - x_1^j - \dots - x_k^j, \quad 1 \le j \le k, \tag{3.2} $$
where $x_1, ..., x_k$ are distinct mod $p$ and $y_1, ..., y_k$ are distinct mod $p$ and since we are considering $J_{s,k}(X, p, 0)$, we have that $x_{k+1}, ..., x_s, y_{k+1}, ..., y_s \equiv 0 \pmod{p}$. Each choice of $x_1, ..., x_k, y_1, ..., y_k$ gives $\le J_{s-k,k}(X/p)$ many solutions to $(x_{k+1}, ..., x_s, y_{k+1}, ..., y_s)$. To see this, write the count for (3.2) as an integral and use the triangle inequality; the basic idea being that shifts of the Vinogradov system can only give fewer solutions.

Next, fixing one of the at most $J_{s-k,k}(X/p)$ many tuples $(x_{k+1}, ..., x_s, y_{k+1}, ..., y_s)$, how many valid $x_1, ..., x_k, y_1, ..., y_k$ are there? As requiring $y_1, ..., y_k$ to be distinct mod $p$ is a rather mild condition, there are $\le X^k$ such $(y_1, ..., y_k)$. Any valid $(x_1, ..., x_k) \in ([1, X] \cap \mathbb{N})^k$ must satisfy

$$x_1^j + \dots + x_k^j \equiv H_j \pmod{p^j}, \quad 1 \le j \le k,$$

where the $x_i$ are pairwise disjoint mod $p$ for some $H_j$ that depends on $(y_1, ..., y_k)$ (of which there are $\le X^k$ many possibilities) and $(x_{k+1}, ..., x_s, y_{k+1}, ..., y_s)$ (of which there are $\le J_{s-k,k}(X/p)$ many possibilities). As $p^k \sim X$, instead of counting integers between 1 and $X$, we can count the $x_i$ mod $p^k$. Thus, it remains to count the number of residue classes $(x_1 \pmod{p^k}, ..., x_k \pmod{p^k})$ such that

$$x_1^j + \dots + x_k^j \equiv H_j \pmod{p^j}, \quad 1 \le j \le k \qquad (3.3)$$

and $x_i \pmod{p^k}$ are pairwise distinct mod $p$. Linnik's lemma [24] then says that there are at most $k!p^{k(k-1)/2}$ many such $k$-tuples of residue classes and the proof follows from first upgrading all residue classes mod $p^j$ in (3.3) to mod $p^k$ (by paying a cost of $p^{k(k-1)/2}$) and then using the Newton-Girard identities that essentially uniquely determine the $x_1, ..., x_k$ (up to permutation). This bound is efficient because probabilistic heuristics suggest that we should expect $\approx (p^k)^k/p^{k(k+1)/2} = p^{k(k-1)/2}$ many solutions. Thus, we have that

$$J_{s,k}(X, p, 0) \lesssim_k J_{s-k,k}(X/p)X^k p^{k(k-1)/2}. \qquad (3.4)$$

## 3.4 | Step 4: Iteration

Putting Steps 1-3 together, we obtain the iteration that

$$J_{s,k}(X) \lesssim_k p^{2s-2k} J_{s-k,k}(X/p) X^k p^{k(k-1)/2}. \qquad (3.5)$$

Running this iteration about $O(s/k)$ many steps reduces to an estimate on $J_{k,k}(X)$ from which one can easily compute there are $O(X^k)$ many solutions by the Newton-Girard identities. The iteration (3.5) is sharp if both $s$ and $s-k$ are supercritical. If they are, then heuristically, we expect $J_{s,k}(X) \approx X^{2s-k(k+1)/2}$ and $J_{s-k,k}(X/p) \approx (X/p)^{2(s-k)-k(k+1)/2}$. Then the right-hand side of (3.5) becomes $X^{2s}X^{-3k/2-k^2/2}p^{k^2}$ which is equal to $X^{2s-k(k+1)/2}$ because $p \sim X^{1/k}$. However, both sides are not the same if one of $s$ or $s-k$ is subcritical. This is where the inefficiency of $X^{k^2/2}(1-1/k)^{s/k}$ comes from.

## 3.5 | Interpreting Steps 1-4 into decoupling

Having briefly summarized the number theoretic argument into four steps, we now briefly sketch the main points to interpret into decoupling. First we discuss the scales needed in the proof.
From Steps 1 and 3, there are three scales: the largest scale $X$, the intermediate scale $p \sim X^{1/k}$, and the smallest scale 1. Correspondingly in our proof, we use three scales: the smallest scale $\delta$, the intermediate scale $\nu := q^{\lfloor \log_q \delta^{1/k} \rfloor} \sim \delta^{1/k}$, and the largest scale 1. For some technical reasons surrounding the broad-narrow reduction, in lieu of the scale 1, we will actually use the scale $\kappa := q^{\lfloor \log_q \delta^\epsilon \rfloor}$ where $\epsilon$ is as in (1.8).

Next, we discuss the reduction to the decoupling analogue of (3.1). In Step 1, two residue classes being distinct mod $p$ means they are $p$-adically separated by a distance 1 and so this should correspond to two intervals that are 1-separated. To get around the use of the Prime Number Theorem, we instead make use of broad-narrow reduction due to Bourgain and Guth in [4] which will allow us to reduce to controlling a multilinear decoupling expression.

Third, the loss of $p^{2s-2k}$ in Step 2 above deserves some mention. This loss comes from essentially having applied the union bound

$$ \left| \sum_{1 \le n \le X} e(n\alpha_1 + \dots + n^k \alpha_k) \right| = \left| \sum_{a \pmod p} \sum_{n \equiv a \pmod p} e(n\alpha_1 + \dots + n^k \alpha_k) \right| \\ \le p \max_{a \pmod p} \left| \sum_{\substack{n \equiv a \pmod p \\ 1 \le n \le X}} e(n\alpha_1 + \dots + n^k \alpha_k) \right|. $$

Heuristically, we expect this inequality to be efficient because each $\sum_{n \equiv a \pmod p}$ contributes equally to the entire sum as the exponential sum should not bias one residue class mod $p$ over another. This, however, is not necessarily true in the decoupling case and will require us to obtain some extra uniformity via dyadic pigeonholing, see Subsection 5.1, later.

Finally, to interpret the solution counting Step 3, we make use of the simple identity

$$ \int_{\mathbb{Q}_q^k} f(x) dx = \hat{f}(0), $$

which converts the integral of $f$ into a question of whether 0 is contained in the support of $\hat{f}$. This is done in Lemma 4.4 and the proof relies on the Newton-Girard identities, much like in the proof of Linnik's lemma. This part of the argument requires that $p$ is even and is reminiscent of a Córdoba-Fefferman argument (see, for example, [8, section 3.2] or [6, 7, 10]).

## 4 | THE MAIN LEMMA

One standard property about the moment curve decoupling constant that we use is affine rescaling. This property plays the analogue of translation-dilation invariance of the Vinogradov system (1.1).

**Lemma 4.1** (Affine rescaling). Let $g$ be a Schwartz function on $\mathbb{Q}_q^k$ Fourier supported in $\bigcup_{K \in P_\delta} \theta_K$. Then for any interval $I \subset \mathbb{Z}_q$ of length $\kappa \ge \delta$, we have

$$ \|g_I\|_{LP(\mathbb{Q}_q^k)} \le \mathfrak{D}_p \left( \frac{\delta}{\kappa} \right) \left( \sum_{K \in P_\delta(I)} \|g_K\|_{LP(\mathbb{Q}_q^k)}^2 \right)^{1/2}. $$
**Proof.** This proof is standard and follows from a change of variables that can be found, for example, in [8, section 11.2]. ☐

Our main lemma in proving Theorem 1.1 is the following:

**Lemma 4.2.** Let $p \in 2k + 2\mathbb{N}$, $\delta \in q^{-\mathbb{N}}$ and $\kappa \in q^{-\mathbb{N}} \cap [\delta, 1)$. Let $\nu = q^{\lfloor \log_q \delta^{1/k} \rfloor} \in q^{-\mathbb{N}}$ so that $\nu \le \delta^{1/k}$. If $g$ is a Schwartz function with Fourier support in $\bigcup_{K \in P_\delta} \theta_K$, then we have

$$ \int_{\mathbb{Q}_q^k} |g|^p \le C \mathfrak{D}_p (\frac{\delta}{\kappa})^p \left( \sum_{K \in P_\delta} \|g_K\|_{LP(\mathbb{Q}_q^k)}^2 \right)^{p/2} + C q^{-k(k-1)} \kappa^{-(k^2+4k-2)} \nu^{-k(k-1)/2} N^{p-2k} \times \\ \mathfrak{D}_{p-2k} (\frac{\delta}{\nu})^{p-2k} \max_{K \in P_\delta} \|g_K\|_{L^\infty(\mathbb{Q}_q^k)}^k \left( \sum_{K' \in P_\delta} \|g_{K'}\|_{L^\infty(\mathbb{Q}_q^k)}^2 \right)^k \max_{J \in P_\nu} \left( \sum_{K' \in P_\delta(J)} \|g_{K'}\|_{L^{p-2k}(\mathbb{Q}_q^k)}^2 \right)^{(p-2k)/2}, $$

where $N$ is the number of $J \in P_\nu$ for which $g_J \ne 0$ and $C$ depends only on $k$ and $p$.

Here $\kappa$ is a somewhat technical parameter that is chosen to be roughly $\delta^\epsilon$ later in Section 5. However, on a first reading, it might be more convenient for the reader to take $\kappa = 1/q$ to better grasp the moving parts of the argument. The somewhat nonstandard decoupling right-hand side in Lemma 4.2 is reminiscent of the right-hand side used in [18, Theorem 1.2]. To give more context to the above lemma, the following estimate is true:

**Lemma 4.3.** For any $p > 2k$, we have

$$ \left( \sum_{K \in P_\delta} \|g_K\|_{LP(\mathbb{Q}_q^k)}^2 \right)^{p/2} \le N^{(p-2k)/2} \max_{K \in P_\delta} \|g_K\|_{L^\infty(\mathbb{Q}_q^k)}^k \left( \sum_{K' \in P_\delta} \|g_{K'}\|_{L^\infty(\mathbb{Q}_q^k)}^2 \right)^k \max_{J \in P_\nu} \left( \sum_{K \in P_\delta(J)} \|g_K\|_{L^{p-2k}(\mathbb{Q}_q^k)}^2 \right)^{(p-2k)/2}, $$

where $N$ is as defined in Lemma 4.2.

**Proof.** Hölder's inequality gives us

$$ \|g_K\|_{LP(\mathbb{Q}_q^k)} \le \|g_K\|_{L^\infty(\mathbb{Q}_q^k)}^{\frac{2k}{p}} \|g_K\|_{L^{p-2k}(\mathbb{Q}_q^k)}^{1-\frac{2k}{p}}, $$

and so, applying

$$ \left( \sum_K a_K^{\frac{2k}{p}} b_K^{\frac{2k}{p}} c_K^{2(1-\frac{2k}{p})} \right)^{\frac{p}{2}} \le (\max_K a_K)^k \left( \sum_K b_K \right)^k \left( \sum_K c_K^2 \right)^{\frac{p-2k}{2}} $$
with $a_K = b_K = \|g_K\|_{L^\infty(\mathbb{Q}_q^k)}$ and $c_K = \|g_K\|_{L^{p-2k}(\mathbb{Q}_q^k)}$, we get

$$ \left( \sum_{K \in P_{\delta}} \|g_K\|_{L^p(\mathbb{Q}_q^k)}^2 \right)^{p/2} \le \max_{K \in P_{\delta}} \|g_K\|_{L^\infty(\mathbb{Q}_q^k)}^k \left( \sum_{\bar{K} \in P_{\delta}} \|g_{\bar{K}}\|_{L^\infty(\mathbb{Q}_q^k)} \right)^k \left( \sum_{K \in P_{\delta}} \|g_K\|_{L^{p-2k}(\mathbb{Q}_q^k)}^2 \right)^{(p-2k)/2}. $$

It remains to observe that

$$ \left( \sum_{K \in P_{\delta}} \|g_K\|_{L^{p-2k}(\mathbb{Q}_q^k)}^2 \right)^{(p-2k)/2} \le N^{(p-2k)/2} \max_{J \in P_{\kappa}} \left( \sum_{K \in P_{\delta}(J)} \|g_K\|_{L^{p-2k}(\mathbb{Q}_q^k)}^2 \right)^{(p-2k)/2}. \quad \square $$

Suppose for a moment that in Lemma 4.3, we had an equality instead of an inequality. This is indeed the case when $g(x)$ is the exponential sum $X^{-100k^2}1_{|x| \le X^{100k}} \sum_{j=1}^X e(\gamma(j) \cdot x)$ that arises in using decoupling to estimate the number of solutions in (1.3). As $N \le \nu^{-1}$ (and taking, for convenience, $\kappa = 1/q$), Lemma 4.2 would give us

$$ \mathfrak{D}_p(\delta)^p \le C \mathfrak{D}_p(q\delta)^p + Cq^{5k-2}\nu^{-\frac{p}{2}+k-\frac{k(k-1)}{2}} \mathfrak{D}_{p-2k}\left(\frac{\delta}{\nu}\right)^{p-2k} \quad (4.1) $$

Heuristically, we expect this iteration to be efficient as long as $p - 2k$ (and so also $p$) is supercritical. To see this, if $r$ is supercritical, then we heuristically expect that $\mathfrak{D}_r(\delta)^r \approx \delta^{-\frac{r}{2}+\frac{k(k+1)}{2}}$ for all $\delta$. Thus, the iteration should be efficient if with this assumption on the size of $\mathfrak{D}_r(\delta)^r$, both sides of (4.1) are the same. The right-hand side of (4.1) is then

$$ \sim_q \left( \delta^{-\frac{1}{k}} \right)^{\frac{p}{2}+k-\frac{k(k-1)}{2}} \left( \delta^{-1+\frac{1}{k}} \right)^{\frac{p-2k}{2}-\frac{k(k+1)}{2}} = \delta^{-\left(\frac{p}{2}-\frac{k(k+1)}{2}\right)}, $$

which is comparable to the left-hand side of (4.1). A similar calculation shows that this iteration is not efficient if at least one of $p$ or $p - 2k$ is subcritical.

Unfortunately the reverse inequality in Lemma 4.3 fails to hold for general $g$. This is because we lack the uniformity in the exponential sum that one considers when one counts solutions to the Vinogradov system. This uniformity can be restored by pigeonholing, which only produces $\delta^{-\epsilon}$ losses. This pigeonholing must be done before one applies induction on scales and iterates on the Lebesgue exponent $p$. The full argument is carried out in detail in Section 5.

## 4.1 | Proof of Lemma 4.2

The proof of Lemma 4.2 uses a broad/narrow dichotomy, due to Bourgain and Guth [4] combined with some basic geometric geometric properties of the moment curve. See also, for example, [8, chapter 7].

### 4.1.1 | The broad-narrow argument

First, we have the pointwise bound $|g(x)| \le \sum_{I \in P_{\kappa}} |g_I(x)|$. At every point $x \in \mathbb{Q}_q^k$, let $I_x$ be the set of all intervals $I' \in P_{\kappa}$ such that $|g_{I'}(x)| \ge \kappa \max_{I \in P_{\kappa}} |g_I(x)|$. Suppose first $I_x$ contains at least $k$
(disjoint) intervals, say $I'_1, \dots, I'_k$ (all dependent on $x$) of length $\kappa$ and $|g_{I'_1}(x)| = \max_{I \in P_\kappa} |g_I(x)|$; in this case we have

$$ 
\begin{aligned}
|g(x)| &\le \kappa^{-1} \max_{I \in P_\kappa} |g_I(x)| \le \kappa^{-1} \kappa^{-(k-1)/k} |g_{I'_1}(x) \dots g_{I'_k}(x)|^{1/k} \\
&\le \kappa^{-2+1/k} \max_{\substack{I_1, \dots, I_k \in P_\kappa \\ d(I_i, I_j) > \kappa \forall i \ne j}} |g_{I_1}(x) \dots g_{I_k}(x)|^{1/k}.
\end{aligned}
$$

Here we used that in $\mathbb{Q}_q^k$, two distinct intervals of the same length are separated by at least that length. Alternatively, $I_x$ contains at most $k-1$ intervals, in which case

$$ |g(x)| \le \sum_{I \in I_x} |g_I(x)| + \sum_{I \in P_\kappa \setminus I_x} |g_I(x)| < (k-1) \max_{I \in P_\kappa} |g_I(x)| + \sum_{I \in P_\kappa \setminus I_x} \kappa \max_{I \in P_\kappa} |g_I(x)| < k \max_{I \in P_\kappa} |g_I(x)|. $$

As a result, we obtain the pointwise bound that for each $x \in \mathbb{Q}_q^k$, we have

$$ |g(x)| \le k \max_{I \in P_\kappa} |g_I(x)| + \kappa^{-2+1/k} \max_{\substack{I_1, \dots, I_k \in P_\kappa \\ d(I_i, I_j) > \kappa \forall i \ne j}} |g_{I_1}(x) \dots g_{I_k}(x)|^{1/k} $$

which, upon raising both sides to power $2k$ and applying $(A+B)^{2k} \le 2^{2k-1}(A^{2k} + B^{2k})$ (a consequence of the convexity of $x \mapsto x^{2k}$), yields

$$ |g(x)|^{2k} \le 2^{2k-1} k^{2k} \max_{I \in P_\kappa} |g_I(x)|^{2k} + 2^{2k-1} \kappa^{-(4k-2)} \max_{\substack{I_1, \dots, I_k \in P_\kappa \\ d(I_i, I_j) > \kappa \forall i \ne j}} |g_{I_1}(x) \dots g_{I_k}(x)|^2. \quad (4.2) $$

Using this pointwise bound while integrating we find that

$$ 
\begin{aligned}
\int_{\mathbb{Q}_q^k} |g|^p &= \int_{\mathbb{Q}_q^k} |g|^{2k} |g|^{p-2k} \\
&\le C \int_{\mathbb{Q}_q^k} (\max_{I \in P_\kappa} |g_I|^2)^k |g|^{p-2k} + C \kappa^{-(4k-2)} \max_{\substack{I_1, \dots, I_k \in P_\kappa \\ d(I_i, I_j) > \kappa \forall i \ne j}} |g_{I_1} \dots g_{I_k}|^2 |g|^{p-2k} \\
&\le C \int_{\mathbb{Q}_q^k} \left( \sum_{I \in P_\kappa} |g_I|^2 \right)^k |g|^{p-2k} + C \kappa^{-(4k-2)} \kappa^{-k} \max_{\substack{I_1, \dots, I_k \in P_\kappa \\ d(I_i, I_j) > \kappa \forall i \ne j}} |g_{I_1} \dots g_{I_k}|^2 |g|^{p-2k}
\end{aligned}
$$
for some C depending on k. Hölder's inequality followed by Minkowski's inequality implies that the first term satisfies

$$
\begin{align*}
C \int_{\mathbb{Q}_q^k} \left( \sum_{I \in P_\kappa} |g_I|^2 \right)^k |g|^{p-2k} &\le C \left( \int_{\mathbb{Q}_q^k} \left( \sum_{I \in P_\kappa} |g_I|^2 \right)^{p/2} \right)^{2k/p} \left( \int_{\mathbb{Q}_q^k} |g|^p \right)^{(p-2k)/p} \\
&\le C \left( \sum_{I \in P_\kappa} \|g_I\|_{L^p(\mathbb{Q}_q^k)}^2 \right)^k \left( \int_{\mathbb{Q}_q^k} |g|^p \right)^{(p-2k)/p} \\
&\le \frac{1}{2} \int_{\mathbb{Q}_q^k} |g|^p + C' \left( \sum_{I \in P_\kappa} \|g_I\|_{L^p(\mathbb{Q}_q^k)}^2 \right)^{p/2}
\end{align*}
$$

for some $C'$ that depends on $k$ and $p$. The last inequality uses Young's inequality and the fact that $p \ge 2k$. Therefore,

$$
\int_{\mathbb{Q}_q^k} |g|^p \lesssim_p \left( \sum_{I \in P_\kappa} \|g_I\|_{L^p(\mathbb{Q}_q^k)}^2 \right)^{p/2} + \kappa^{-(5k-2)} \max_{\substack{I_1, \dots, I_k \in P_\kappa \\ d(I_i, I_j) > \kappa \forall i \neq j}} \int_{\mathbb{Q}_q^k} |g_{I_1} \dots g_{I_k}|^2 |g|^{p-2k}.
$$

Using affine rescaling (Lemma 4.1) and applying the definition (1.5) of our decoupling constant,
we deduce that

$$
\left( \sum_{I \in P_{\kappa}} \|g_I\|_{L^p(\mathbb{Q}_q^k)}^2 \right)^{p/2} \le \mathfrak{D}_p \left( \frac{\delta}{\kappa} \right)^p \left( \sum_{K \in P_{\delta}} \|g_K\|_{L^p(\mathbb{Q}_q^k)}^2 \right)^{p/2} .
$$

Plugging this into the above yields

$$
\int_{\mathbb{Q}_q^k} |g|^p \lesssim_p \mathfrak{D}_p (\frac{\delta}{\kappa})^p \left( \sum_{K \in P_\delta} \|g_K\|_{L^p(\mathbb{Q}_q^k)}^2 \right)^{p/2} + \kappa^{-(5k-2)} \max_{\substack{I_1, \dots, I_k \in P_\kappa \\ d(I_i, I_j) > \kappa \ \forall i \neq j}} \int_{\mathbb{Q}_q^k} |g_{I_1} \dots g_{I_k}|^2 |g|^{p-2k}. \quad (4.3)
$$

This inequality (4.3) is the analogue of Step 1 in Subsection 3.1. The requirement that we analyze
solutions to the Vinogradov system with $x_1, \dots, x_s$ and $y_1, \dots, y_s$ being distinct mod $p$ corresponds
to the requirement that we analyze $\int_{\mathbb{Q}_q^k} |g_{I_1} \dots g_{I_k}|^2 |g|^{p-2k}$ with $d(I_i, I_j) > \kappa$ for all $1 \le i \ne j \le k$
with $\kappa = 1/q$.

Next, we mimic Step 2 in Subsection 3.2. Recalling our definition of N in the statement of
Lemma 4.2, Hölder's inequality gives

$$
|g|^{p-2k} \le N^{p-2k-1} \sum_{J \in P_v} |g_J|^{p-2k}.
$$

Applying this in the second term in (4.3), we get
$$
\begin{align}
& \int_{\mathbb{Q}_q^k} |g|^p \nonumber \\
& \lesssim_p \mathfrak{D}_p \left(\frac{\delta}{\kappa}\right)^p \left(\sum_{K \in P_\delta} \|g_K\|_{L^P(\mathbb{Q}_q^k)}^2\right)^{p/2} + \kappa^{-(5k-2)} N^{p-2k-1} \max_{\substack{I_1, \dots, I_k \in P_\kappa \\ d(I_i, I_j) > \kappa \forall i \neq j}} \sum_{J \in P_\nu} \int_{\mathbb{Q}_q^k} |g_{I_1} \dots g_{I_k}|^2 |g_J|^{p-2k} \nonumber \\
& \lesssim_p \mathfrak{D}_p \left(\frac{\delta}{\kappa}\right)^p \left(\sum_{K \in P_\delta} \|g_K\|_{L^P(\mathbb{Q}_q^k)}^2\right)^{p/2} + \kappa^{-(5k-2)} N^{p-2k} \max_{\substack{I_1, \dots, I_k \in P_\kappa \\ d(I_i, I_j) > \kappa \forall i \neq j}} \max_{J \in P_\nu} \int_{\mathbb{Q}_q^k} |g_{I_1} \dots g_{I_k}|^2 |g_J|^{p-2k} \tag{4.4}
\end{align}
$$

which is the analogue of Step 2 in Subsection 3.2.

To analyze the second term in (4.4), we fix $I_1, \dots, I_k \in P_\kappa$ with $d(I_i, I_j) > \kappa$ for all $i \neq j$, and fix $J \in P_\nu$ with $g_J \neq 0$. To estimate the integral $\int_{\mathbb{Q}_q^k} |g_{I_1} \dots g_{I_k}|^2 |g_J|^{p-2k}$, first note that the Fourier transform of $|g_J|^2 = g_J \overline{g_J}$ is supported in the parallelepiped $\theta_J - \theta_J$, of dimension $\nu \times \nu^2 \times \dots \times \nu^k$. As our hypothesis guarantees that $p-2k$ is an even positive integer, the same is true for the Fourier transform of $|g_J|^{p-2k}$. Lemma 2.1(i) applied to $J \in P_\nu$ instead of $K \in P_\delta$ shows that the Fourier support of $|g_J|^{p-2k}$ is the disjoint union of $\nu^{-k(k-1)/2}$ many cubes of side lengths $\nu^k$, and we denote this collection of cubes by $\{\square\}$. This corresponds to the fact that we have a $k$-tuple of residue classes $(H_1 \pmod p), H_2 \pmod{p^2}), \dots, H_k \pmod{p^k})$ that we can upgrade to $p^{k(k-1)/2}$ many $k$-tuples of the form $(H'_1 \pmod{p^k}), H'_2 \pmod{p^k}), \dots, H'_k \pmod{p^k})$. Note that the side length $\nu^k$ of the cubes $\square$ is $\le \delta$.

We now apply Fourier inversion and turn products into convolutions. We have

$$
\begin{align*}
\int_{\mathbb{Q}_q^k} |g_{I_1} \dots g_{I_k}|^2 |g_J|^{p-2k} &= \sum_{K_i \in P_\delta(I_i)} \sum_{\bar{K}_j \in P_\delta(I_j)} \int_{\mathbb{Q}_q^k} g_{K_1} \dots g_{K_k} \overline{g_{\bar{K}_1}} \dots \overline{g_{\bar{K}_k}} |g_J|^{p-2k} \\
&= \sum_{\square} \sum_{K_i \in P_\delta(I_i)} \sum_{\bar{K}_j \in P_\delta(I_j)} \widehat{g_{K_1}} * \dots * \widehat{g_{K_k}} * \widehat{g_{\bar{K}_1}} * \dots * \widehat{g_{\bar{K}_k}} * (|g_J|^{p-2k} 1_{\square})(0).
\end{align*}
$$

For each fixed $\square$ and $\bar{K}_1 \in P_\delta(I_1), \dots, \bar{K}_k \in P_\delta(I_k)$, let $S(\bar{K}_1, \dots, \bar{K}_k, \square)$ be the set of all $(K_1, \dots, K_k)$ with $K_i \in P_\delta(I_i)$ such that

$$
0 \in \mathrm{supp}(\widehat{g_{K_1}} * \dots * \widehat{g_{K_k}} * \widehat{g_{\bar{K}_1}} * \dots * \widehat{g_{\bar{K}_k}} * (|g_J|^{p-2k} 1_{\square})). \quad (4.5)
$$

We will prove in Lemma 4.4 that $\#S(\bar{K}_1, \dots, \bar{K}_k, \square) \le (q\kappa)^{-k(k-1)}$. If we think of the model case when $\kappa = 1/q$, this would say that the $\bar{K}_i$ and $\square$ uniquely determine the $K_i$ in (4.5). This is analogous to the situation in Linnik's lemma where once we upgrade (3.3) to residue classes mod $p^k$, the remaining variables are essentially uniquely determined.

We now write

$$
\int_{\mathbb{Q}_q^k} |g_{I_1} \dots g_{I_k}|^2 |g_J|^{p-2k} \\
= \left| \sum_{\square} \sum_{\bar{K}_j \in P_\delta(I_j)} \sum_{\substack{(K_1, \dots, K_k) \\ j=1, \dots, k}} \widehat{g_{K_1}} * \dots * \widehat{g_{K_k}} * \widehat{g_{\bar{K}_1}} * \dots * \widehat{g_{\bar{K}_k}} * (|g_J|^{p-2k} 1_{\square}) (0) \right|
$$
$$ \le \sum_{\square} \sum_{\bar{K}_j \in P_\delta(I_j)} \sum_{\substack{K_1, \dots, K_k \\ j=1, \dots, k}} \sum_{S(\bar{K}_1, \dots, \bar{K}_k, \square)} \int_{\mathbb{Q}_q^k} |g_{\bar{K}_1} \dots g_{\bar{K}_k} g_{\bar{K}_1} \dots g_{\bar{K}_k}||g_J|^{p-2k} * |\widetilde{1}_{\square}| $$

$$ \le \sum_{\square} \left( \sum_{\bar{K} \in P_\delta} \|g_K\|_{L^\infty(\mathbb{Q}_q^k)} \right)^k (q\kappa)^{-k(k-1)} \max_{K \in P_\delta} \|g_K\|_{L^\infty(\mathbb{Q}_q^k)}^k \int_{\mathbb{Q}_q^k} |g_J|^{p-2k} * |\widetilde{1}_{\square}|. $$

As $\int_{\mathbb{Q}_q^k} |\widetilde{1}_{\square}| = 1$ and the number of $\square$ is $\nu^{-k(k-1)/2}$, this gives

$$ \int_{\mathbb{Q}_q^k} |g_{I_1} \dots g_{I_k}|^2 |g_J|^{p-2k} \le \nu^{-\frac{k(k-1)}{2}} (q\kappa)^{-k(k-1)} \max_{\bar{K} \in P_\delta} \|g_K\|_{L^\infty(\mathbb{Q}_q^k)}^k \left( \sum_{\bar{K} \in P_\delta} \|g_{\bar{K}}\|_{L^\infty(\mathbb{Q}_q^k)} \right)^k \int_{\mathbb{Q}_q^k} |g_J|^{p-2k}. $$

Applying affine rescaling shows that this is

$$ \le \nu^{-\frac{k(k-1)}{2}} (q\kappa)^{-k(k-1)} \max_{K \in P_\delta} \|g_K\|_{L^\infty(\mathbb{Q}_q^k)}^k \times \left( \sum_{\bar{K} \in P_\delta} \|g_K\|_{L^\infty(\mathbb{Q}_q^k)} \right)^k \mathfrak{D}_{p-2k} \left(\frac{\delta}{\nu}\right)^{p-2k} \left( \sum_{K' \in P_\delta(J)} \|g_{K'}\|_{L^{p-2k}(\mathbb{Q}_q^k)}^2 \right)^{\frac{p-2k}{2}}. \quad (4.6) $$

One can think of (4.6) as the analogue of (3.4) in Subsection 3.3 in the following way: the term $\nu^{-k(k-1)/2}(q\kappa)^{-k(k-1)} \max_{K \in P_\delta} \|g_K\|_\infty^k$ plays the role of $p^{k(k-1)/2}$ from Linnik's lemma, the term $(\sum_{\bar{K} \in P_\delta} \|g_K\|_\infty)^k$ plays the role of $X^k$, and finally the term $\mathfrak{D}_{p-2k}(\frac{\delta}{\nu})^{p-2k} (\sum_{K' \in P_\delta(J)} \|g_{K'}\|_{L^\infty(\mathbb{Q}_q^k)}^2)^{\frac{p-2k}{2}}$ plays the role of the $J_{s-k,k}(X/p)$.

Plugging (4.6) back to (4.4), we then obtain

$$ \int_{\mathbb{Q}_q^k} |g|^p \lesssim_p \mathfrak{D}_p (\frac{\delta}{\kappa})^p \left( \sum_{K \in P_\delta} \|g_K\|_{L^p(\mathbb{Q}_q^k)}^2 \right)^{p/2} + q^{-k(k-1)} \kappa^{-(k^2+4k-2)} \nu^{-\frac{k(k-1)}{2}} N^{p-2k} \times \\ \mathfrak{D}_{p-2k} (\frac{\delta}{\nu})^{p-2k} \max_{K \in P_\delta} \|g_K\|_{L^\infty(\mathbb{Q}_q^k)}^k \left( \sum_{\bar{K} \in P_\delta} \|g_{\bar{K}}\|_{L^\infty(\mathbb{Q}_q^k)} \right)^k \left( \sum_{K' \in P_\delta(J)} \|g_{K'}\|_{L^{p-2k}(\mathbb{Q}_q^k)}^2 \right)^{\frac{p-2k}{2}}. $$

### 4.1.2 | Geometry of the moment curve

The proof of Lemma 4.2 is now complete modulo the proof of the following lemma, which provides the key geometric input that enables one to count $\#S(\bar{K}_1, \dots, \bar{K}_k, \square)$. This is the analogue of Linnik's lemma ([29, Corollary 17] and the estimate for $\mathbf{B}(g)$ in the proof of [31, Lemma 5.1]); see also [13, Proposition 1.3] and [2, Proposition 3.1]. Both proofs use the Newton-Girard identities in essentially the same way. The hypothesis that $q > k$, where $q$ is from our base field $\mathbb{Q}_q$ and $k$ is the degree of the moment curve, plays a role in the following lemma.
**Lemma 4.4.** Let $p \in 2k + 2\mathbb{N}$, $\delta \in q^{-\mathbb{N}}$, $\kappa \in q^{-\mathbb{N}} \cap [\delta, 1)$, and $\nu = q^{\lfloor \log_q \delta^{1/k} \rfloor} \in q^{-\mathbb{N}}$ so that $\nu \le \delta^{1/k}$. Suppose that $I_1, \dots, I_k \in P_\kappa$ with $d(I_i, I_j) > \kappa$ for all $i \neq j$. Let $\Box$ be a cube of side length $\nu^k$ and $\bar{K}_1 \in P_\delta(I_1), \dots, \bar{K}_k \in P_\delta(I_k)$. Define $S(\bar{K}_1, \dots, \bar{K}_k, \Box)$ be the set of all ordered $k$-tuples $(K_1, \dots, K_k)$ with $K_i \in P_\delta(I_i)$ such that

$$0 \in \mathrm{supp}(\widehat{g}_{\bar{K}_1} * \dots * \widehat{g}_{\bar{K}_k} * \widehat{g}_{\bar{K}_1} * \dots * \widehat{g}_{\bar{K}_k} * (|g_J|^{p-2k} 1_{\Box})).$$

Then

$$\#S(\bar{K}_1, \dots, \bar{K}_k, \Box) \le (q\kappa)^{-k(k-1)}.$$

*Proof.* Assume for the sake of contradiction that $\#S(\bar{K}_1, \dots, \bar{K}_k, \Box) > (q\kappa)^{-k(k-1)} \ge 1$. We can find two $k$-tuples of intervals $(A_1, \dots, A_k)$ and $(B_1, \dots, B_k)$ with each $A_i, B_i \in P_\delta(I_i)$ such that

$$0 \in \mathrm{supp}(\widehat{g}_{A_1} * \dots * \widehat{g}_{A_k} * \widehat{g}_{\bar{K}_1} * \dots * \widehat{g}_{\bar{K}_k} * (|g_J|^{p-2k} 1_{\Box})), \quad (4.7)$$

$$0 \in \mathrm{supp}(\widehat{g}_{B_1} * \dots * \widehat{g}_{B_k} * \widehat{g}_{\bar{K}_1} * \dots * \widehat{g}_{\bar{K}_k} * (|g_J|^{p-2k} 1_{\Box})), \quad (4.8)$$

and such that there exists an $i_0$ with $d(A_{i_0}, B_{i_0}) > (q\kappa)^{-(k-1)}\delta$. Indeed, if not, picking an arbitrary $(C_1, \dots, C_k) \in S(\bar{K}_1, \dots, \bar{K}_k, \Box)$, shows that any other $(D_1, \dots, D_k) \in S(\bar{K}_1, \dots, \bar{K}_k, \Box)$ must satisfy $d(C_i, D_i) \le (q\kappa)^{-(k-1)}\delta$. This gives at most $(q\kappa)^{-k(k-1)}$ many $k$-tuples which violates our initial assumption that $\#S(\bar{K}_1, \dots, \bar{K}_k, \Box) > (q\kappa)^{-k(k-1)}$. Without loss of generality, we may assume that $i_0 = 1$.

As for each $i = 1, 2, \dots, k$, we have $A_i, B_i \subset I_i$ and $d(I_i, I_j) > \kappa$ for all $i \neq j$, this implies

$$d(A_i, A_j) \ge q\kappa, \quad d(B_i, B_j) \ge q\kappa, \quad d(A_i, B_j) \ge q\kappa \quad \text{whenever } j \neq i \quad (4.9)$$

(thus the only distances we do not have any control over are the ones of the form $d(A_i, B_i)$, $i \neq 1$). By (4.7) and (4.8), we have that

$$0 \in \left( \sum_{i=1}^{k} \tau_{A_i} - \sum_{i=1}^{k} \tau_{\bar{K}_i} + \Box \right) \cap \left( \sum_{i=1}^{k} \tau_{B_i} - \sum_{i=1}^{k} \tau_{\bar{K}_i} + \Box \right)$$

where here we recall the definition of $\tau_K$ in (1.4). Each $\tau_{A_i}, \tau_{B_i}$, and $\tau_{\bar{K}_i}$ are cubes in $\mathbb{Q}_q^k$ of side length $\delta$ and $\Box$ is a cube in $\mathbb{Q}_q^k$ of side length $\nu^k \le \delta$. Thus by the ultrametric inequality, both $\sum_{i=1}^k \tau_{A_i} - \sum_{i=1}^k \tau_{\bar{K}_i} + \Box$ and $\sum_{i=1}^k \tau_{B_i} - \sum_{i=1}^k \tau_{\bar{K}_i} + \Box$ are cubes in $\mathbb{Q}_q^k$ of side length $\delta$. Furthermore, by the ultrametric inequality, as two cubes of side length $\delta$ are either completely disjoint or exactly the same, we must have

$$\sum_{i=1}^{k} \tau_{A_i} - \sum_{i=1}^{k} \tau_{\bar{K}_i} + \Box = \sum_{i=1}^{k} \tau_{B_i} - \sum_{i=1}^{k} \tau_{\bar{K}_i} + \Box$$

and hence

$$\sum_{i=1}^{k} \tau_{A_i} - \sum_{i=1}^{k} \tau_{B_i} = B(0, \delta).$$
Therefore, (after another application of the ultrametric inequality) there exists $\xi_{A_i} \in A_i$ and $\xi_{B_i} \in B_i$ such that

$$ \left| \sum_{i=1}^{k} \xi_{A_i}^{j} - \sum_{i=1}^{k} \xi_{B_i}^{j} \right| \le \delta \qquad (4.10) $$

for $j = 1, 2, ..., k$.

We now use the Newton-Girard identities to derive a contradiction. For $j = 1, 2, ..., k$, define the power sums $p_j(x_1, ..., x_k) := x_1^j + ... + x_k^j$. Next for $j = 1, 2, ..., k$, define the elementary symmetric polynomials $e_j(x_1, ..., x_k) := \sum_{1 \le i_1 < ... < i_j \le k} x_{i_1} ... x_{i_j}$. Additionally, let $e_0(x_1, ..., x_k) := 1$. Then we have the two identities:

$$ (X - x_1)(X - x_2) \cdots (X - x_k) = \sum_{j=0}^{k} (-1)^j e_j(x_1, \dots, x_k) X^{k-j} \qquad (4.11) $$

and for $j = 1, 2, ..., k$, we have

$$ je_j(x_1, \dots, x_k) = \sum_{i=0}^{j-1} (-1)^i e_{j-i-1}(x_1, \dots, x_k) p_{i+1}(x_1, \dots, x_k). \qquad (4.12) $$

See, for example, [29, Lemma 15] for a proof.

Let $e_j(A) := e_j(\xi_{A_1}, ..., \xi_{A_k})$ and $p_j(A) := p_j(\xi_{A_1}, ..., \xi_{A_k})$. Similarly define $e_j(B)$ and $p_j(B)$. By (4.11), we then have

$$ (\xi_{A_1} - \xi_{B_1}) \cdots (\xi_{A_k} - \xi_{B_k}) = \sum_{j=0}^{k} (-1)^j e_j(B) \xi_{A_1}^{k-j} \qquad (4.13) $$

and

$$ 0 = (\xi_{A_1} - \xi_{A_1}) \cdots (\xi_{A_k} - \xi_{A_k}) = \sum_{j=0}^{k} (-1)^j e_j(A) \xi_{A_1}^{k-j}. \qquad (4.14) $$

Subtracting (4.14) from (4.13) and using that $|\xi_{A_1} - \xi_{B_j}| \ge q\kappa$ for any $j \ne 1$ (which follows from (4.9)) shows that

$$ (q\kappa)^{k-1} |\xi_{A_1} - \xi_{B_1}| \le |\sum_{j=0}^{k} (-1)^j (e_j(B) - e_j(A)) \xi_{A_1}^{k-j}| \le \max_j |e_j(B) - e_j(A)|. \qquad (4.15) $$

Next we claim that $|e_j(B) - e_j(A)| \le \delta$ for all $j = 1, 2, ..., k$. We prove this by induction. As $e_1 = p_1$, $|e_1(B) - e_1(A)| \le \delta$ by the $j = 1$ case of (4.10). Now assume that for some $J = 1, 2, ..., k-1$ we had $|e_j(B) - e_j(A)| \le \delta$ for all $j = 1, 2, ..., J$. Then by (4.12),

$$ \begin{aligned} |(J+1)e_{J+1}(B) - (J+1)e_{J+1}(A)| &= |\sum_{i=0}^{J} (-1)^i (e_{J-i}(B) p_{i+1}(B) - e_{J-i}(A) p_{i+1}(A))| \\ &\le \max_{0 \le i \le J} |e_{J-i}(B) p_{i+1}(B) - e_{J-i}(A) p_{i+1}(A)|. \end{aligned} $$
Observe that

$$|e_{J-i}(B)p_{i+1}(B) - e_{J-i}(A)p_{i+1}(A)| = |e_{J-i}(B)(p_{i+1}(B) - p_{i+1}(A)) + p_{i+1}(A)(e_{J-i}(B) - e_{J-i}(A))| \\ \le \max(|p_{i+1}(B) - p_{i+1}(A)|, |e_{J-i}(B) - e_{J-i}(A)|) \le \delta$$

by the inductive hypothesis and (4.10). As $q$ is a prime $> k$, it follows that $|e_{J+1}(B) - e_{J+1}(A)| \le \delta$. Applying this conclusion to (4.15) then yields that $(q\kappa)^{k-1}|\xi_{A_1} - \xi_{B_1}| \le \delta$. But this contradicts the fact that $d(A_1, B_1) > (q\kappa)^{-(k-1)}\delta$. Therefore, we must have $\#S(\bar{K}_1, ..., \bar{K}_k, \square) \le (q\kappa)^{-k(k-1)}$ which completes the proof of the lemma. □

# 5 | PROOF OF THEOREM 1.1 AND COROLLARY 1.2

## 5.1 | Dyadic pigeonholing

It is more convenient to bound

$$D_p(\delta) := \sup_{\delta_0 \in q^{-\mathbb{N}} \cap [\delta, 1]} \mathfrak{D}_p(\delta_0) \qquad (5.1)$$

instead of $\mathfrak{D}_p(\delta)$ as $D_p(\delta)$ is defined for all real $\delta \in (0, 1]$ (rather than just for $\delta \in q^{-\mathbb{N}}$) and is monotonic, that is, $D_p(\delta_L) \le D_p(\delta_S)$ if $\delta_L \ge \delta_S$.

**Proposition 5.1.** For even integers $p > 2k$, there exists a constant $C > 0$, depending only on $k$ and $p$, such that for every $0 < \varepsilon < 1$, we have

$$D_p(\delta)^p \le C(\log \delta^{-1})^{3p} \left[ D_p(\delta^{1-\varepsilon})^p + q^{\frac{p}{2} + \frac{k^2+7k-4}{2}} \delta^{-(k^2+4k-2)\varepsilon} \delta^{-\frac{1}{k}(\frac{p}{2} + \frac{k(k-3)}{2})} D_{p-2k}(\delta^{1-\frac{1}{k}})^{p-2k} \right] \qquad (5.2)$$

for all $0 < \delta < 1$.

*Proof.* To bound $D_p(\delta)^p$, suppose $0 < \delta < 1$ and $\delta_0 \in q^Z$ with $\delta_0 \in [\delta, 1]$. We need to bound $\mathfrak{D}_p(\delta_0)^p$ by decoupling down to frequency scale $\delta_0$.

Let $f$ be a Schwartz function on $\mathbb{Q}_q^k$ with Fourier support in $\cup_{K \in P_{\delta_0}} \theta_K$. Then $f = \sum_{K \in P_{\delta_0}} f_K$ where $\widehat{f}_K := \widehat{f}_{1_{K \times \mathbb{Q}_q^{k-1}}}$. We want to prove the existence of $C > 0$ so that for any $0 < \varepsilon < 1$,

$$\int_{\mathbb{Q}_q^k} |f|^p \le C(\log \delta^{-1})^{3p} \times \left[ D_p(\delta^{1-\varepsilon})^p + q^{\frac{p}{2} + \frac{k^2+7k-4}{2}} \delta^{-(k^2+4k-2)\varepsilon} \delta^{-\frac{1}{k}(\frac{p}{2} + \frac{k(k-3)}{2})} D_{p-2k}(\delta^{1-\frac{1}{k}})^{p-2k} \right] \left( \sum_{K \in P_{\delta_0}} \|f_K\|_{LP(\mathbb{Q}_q^k)}^2 \right)^{p/2} \qquad (5.3)$$
In fact, we will prove that for any translate $Q$ of $B_{\delta_0^{-k}} := \{x \in \mathbb{Q}_q^k : |x| \le \delta_0^{-k}\}$, we have

$$ \int_Q |f|^p \le C(\log \delta^{-1})^{3p} \times \left[ D_p(\delta^{1-\varepsilon})^p + q^{\frac{p}{2} + \frac{k^2+7k-4}{2}} \delta^{-(k^2+4k-2)\varepsilon} \delta^{-\frac{1}{k}(\frac{p}{2} + \frac{k(k-3)}{2})} D_{p-2k}(\delta^{1-\frac{1}{k}})^{p-2k} \right] \left( \sum_{K \in P_{\delta_0}} \|f_K\|_{LP(Q)}^2 \right)^{p/2} . \quad (5.4) $$

The estimate (5.3) then follows by summing over all such $Q$'s that tile $\mathbb{Q}_q^k$, and applying Minkowski's inequality to bring an $\ell^{p/2}$ norm over $Q$ on the right-hand side into the sum over $K \in P_{\delta_0}$.

Thus, we now turn to the proof of (5.4). Note that for any translate $Q$ of $B_{\delta_0^{-k}}$, we have that $\hat{1}_Q$ is supported in $B_{\delta_0^{k-}}$. Therefore, $f1_Q$ is still Fourier supported in $\cup_{K \in P_{\delta_0}} \theta_K$ because $\theta_K + B_{\delta_0^k} = \theta_K$ for all $K \in P_{\delta_0}$. Next, we have $(f1_Q)_K = f_K 1_Q$; indeed

$$ (\hat{f1}_Q)_K = \hat{f1}_Q 1_{K \times \mathbb{Q}_q^{k-1}} = (\hat{f} * \hat{1}_Q) 1_{K \times \mathbb{Q}_q^{k-1}} = (\hat{f1}_{K \times \mathbb{Q}_q^{k-1}}) * \hat{1}_Q = \hat{f}_K * \hat{1}_Q. $$

As a result, to prove (5.4), it suffices to prove (5.3) under the additional assumption that $f$ is supported on $Q$. As $f$ is an arbitrary Schwartz function with Fourier support in $\cup_{K \in P_{\delta_0}} \theta_K$, we may assume $Q = B_{\delta_0^{-k}}$. Thus from now on, we assume additionally that $f$ and all the $f_K$ are supported on $B_{\delta_0^{-k}}$ and prove (5.3).

We first dyadically pigeonhole $f$ by wavepacket height. Write $H^* = \max_{K \in P_{\delta_0}} \|f_K\|_{L^\infty(\mathbb{Q}_q^k)}$. For $K \in P_{\delta_0}$ and $H \in 2^\mathbb{Z}H^* \cap (\delta_0^{1+\frac{k(k-1)}{2p}} H^*, H^*)$, let

$$ f_K^{(H)} = f_K 1_{H/2<|f_K|\le H} $$

where here $f_K : \mathbb{Q}_q^k \to \mathbb{C}$, $|f_K|$ is the absolute value of $f_K$, and the last characteristic function is meant to be the indicator function of the set $\{x \in \mathbb{Q}_q^k : H/2 < |f_K(x)| \le H\}$. As $f_K$ is supported on $B_{\delta_0^{-k}}$, so is $f_K^{(H)}$. By Lemma 2.2, as $f_K$ is Fourier supported in $\theta_K$, we then have

$$ f_K^{(H)} = \left( \sum_{T \in \mathbb{T}(K)} f_K 1_T \right) 1_{H/2 < |f_K| \le H} = \sum_{T \in \mathbb{T}(K)} (f_K 1_T) 1_{H/2 < |f_K 1_T| \le H} \quad (5.5) $$

where the last equality is because $|f_K 1_T|$ constant on every $T \in \mathbb{T}(K)$. Again by Lemma 2.2, note that $f_K^{(H)}$ is Fourier supported in $\theta_K$. Using the terminology of Lemma 2.2, the nonzero wavepackets that make up $f_K^{(H)}$ are all of height $\sim H$. Then

$$ \left\| f - \sum_{K \in P_{\delta_0}} \sum_{H \in 2^{\mathbb{Z}} H^* \cap (\delta_0^{1+\frac{k(k-1)}{2p}} H^*, H^*)} f_K^{(H)} \right\|_{L^\infty(\mathbb{Q}_q^k)} \le \sum_{K \in P_{\delta_0}} \left\| f_K 1_{|f_K| \le \delta_0^{1+\frac{k(k-1)}{2p}} H^*} \right\|_{L^\infty(\mathbb{Q}_q^k)} \\ \le \delta_0^{-1} (\delta_0^{1+\frac{k(k-1)}{2p}} H^*) = \delta_0^{\frac{k(k-1)}{2p}} H^* $$
so as $f$ and $f_K^{(H)}$ are supported on $B_{\delta_0^{-k}}$,

$$ \|f - \sum_{K \in P_{\delta_0}} \sum_{H \in 2^\mathbb{Z} H^* \cap (\delta_0^{1+\frac{k(k-1)}{2p}}, \delta_0^{H^*, H^*}]} f_K^{(H)}\|_{L^p(\mathbb{Q}_q^k)} \le (\delta_0^{\frac{k(k-1)}{2p}} H^*) |B_{\delta_0^{-k}}|^{\frac{1}{p}} = H^* \delta_0^{-\frac{k(k+1)}{2p}} $$

$$ \le \max_{K \in P_{\delta_0}} \|f_K\|_{L^p(\mathbb{Q}_q^k)} \le \left( \sum_{K \in P_{\delta_0}} \|f_K\|_{L^p(\mathbb{Q}_q^k)}^2 \right)^{1/2} $$

where the second inequality follows from writing $f_K = f_K * \tilde{1}_{\theta_K}$ and applying Young's inequality $\|f_K\|_{L^\infty(\mathbb{Q}_q^k)} \le \|f_K\|_{L^p(\mathbb{Q}_q^k)} \|\tilde{1}_{\theta_K}\|_{L^p'(\mathbb{Q}_q^k)}$. This shows

$$ \|f\|_{L^p(\mathbb{Q}_q^k)} \le \sum_{H \in 2^\mathbb{Z} H^* \cap (\delta_0^{1+\frac{k(k-1)}{2p}}, \delta_0^{H^*, H^*}]} \|\sum_{K \in P_{\delta_0}} f_K^{(H)}\|_{L^p(\mathbb{Q}_q^k)} + \left( \sum_{K \in P_{\delta_0}} \|f_K\|_{L^p(\mathbb{Q}_q^k)}^2 \right)^{1/2} $$

Next we dyadically pigeonhole so that each relevant $f_K^{(H)}$ is made up of about the same number of wavepackets. Let now $\nu = q^{\lfloor \log_q \delta_0^{1/k} \rfloor} \le \delta_0^{1/k}$. From (5.5), $f_K^{(H)}$ is Fourier supported in $\theta_K$ and supported in $B_{\delta_0^{-k}}$. As a $T \in \mathbb{T}(K)$ is either completely contained in or completely disjoint from $B_{\delta_0^{-k}}$, we then can write

$$ f_K^{(H)} = \sum_{T \in \mathbb{T}(K), T \subset B_{\delta_0^{-k}}} (f_K 1_T) 1_{H/2 < |f_K 1_T| \le H} \quad (5.6) $$

Furthermore, the $T \in \mathbb{T}(K)$ that are contained in $B_{\delta_0^{-k}}$ perfectly partition $B_{\delta_0^{-k}}$ into $\delta_0^{-k(k-1)/2}$ many translates of $T_{0,k}$. Thus, (5.6) has at most $\delta_0^{-k(k-1)/2}$ many nonzero terms. Therefore for $\alpha \in 2^\mathbb{N} \cap [1, \delta_0^{-k(k-1)/2}]$, let

$$ f_K^{(H, \alpha)} := f_K^{(H)} $$

if the number of nonzero terms in (5.6) (that is, the number of nonzero wavepackets in $f_K^{(H)}$) is in $(\alpha/2, \alpha]$, and 0 otherwise. Thus, now we have that

$$ f_K^{(H)} = \sum_{\alpha \in 2^\mathbb{N} \cap [1, \delta_0^{-k(k-1)/2}]} f_K^{(H, \alpha)} \quad (5.7) $$

and each $f_K^{(H, \alpha)}$ is a function that is supported in $B_{\delta_0^{-k}}$, Fourier supported in $\theta_K$, and has $\sim \alpha$ many nonzero wavepackets of height $\sim H$.

Finally, we dyadically pigeonhole so that given a $K$, the parent interval $J$ of length $\nu$ has about the same number of children $K'$ of length $\delta_0$ such that $f_{K'}^{(H, \alpha)} \ne 0$. To be more precise, fix a $K$ and let $J$ be the unique parent interval of length $\nu$ containing $K$. This parent $J$ contains $\nu/\delta_0$ many intervals $K'$ of length $\delta_0$ and hence $J$ has at most $\nu/\delta_0$ many children $K'$ such that $f_{K'}^{(H, \alpha)} \ne 0$. For
$K \subset J$ and $\beta \in 2^N \cap [1, \nu/\delta_0]$, let

$$f_K^{(H, \alpha, \beta)} := f_K^{(H, \alpha)}$$

if the number of children $K''$ of $J$ with $f_{K''}^{(H, \alpha)} \neq 0$ is in $(\beta/2, \beta]$, that is, if $\#\{K'' \in P_{\delta_0}(J) : f_{K''}^{(H, \alpha)} \neq 0\} \in (\beta/2, \beta]$, and 0 otherwise. Thus, we now have

$$f_K^{(H, \alpha)} = \sum_{\beta \in 2^N \cap [1, \nu/\delta_0]} f_K^{(H, \alpha, \beta)} \quad (5.8)$$

and each $f_K^{(H, \alpha, \beta)}$ is a function that is supported in $B_{\delta_0^{-k}}$, Fourier supported in $\theta_K$, has $\sim \alpha$ many nonzero wavepackets of height $\sim H$, and $K$'s parent $J$ has $\sim \beta$ children each of which also are supported in $B_{\delta_0^{-k}}$, Fourier supported in $\theta_K$, and have $\sim \alpha$ many nonzero wavepackets of height $\sim H$.

Thus, combining (5.7) and (5.8) gives

$$f_K^{(H)} = \sum_{\alpha \in 2^N \cap [1, \delta_0^{-k(k-1)/2}]} \sum_{\beta \in 2^N \cap [1, \nu/\delta_0]} f_K^{(H, \alpha, \beta)},$$

which implies

$$\|f\|_{L^p(\mathbb{Q}_q^k)} \le \sum_{H \in 2^N H^* \cap (\delta_0^{1-k(k-1)/2})} \sum_{\alpha \in 2^N \cap [1, \delta_0^{-k(k-1)/2}]} \sum_{\beta \in 2^N \cap [1, \nu/\delta_0]} \| \sum_{K \in P_{\delta_0}} f_K^{(H, \alpha, \beta)} \|_{L^p(\mathbb{Q}_q^k)} \\ + \left( \sum_{K \in P_{\delta_0}} \| f_K \|_{L^p(\mathbb{Q}_q^k)}^2 \right)^{1/2}.$$

Fix now $\varepsilon > 0$. For each of the $\lesssim (\log \delta_0^{-1})^3$ choices of $(H, \alpha, \beta)$, we apply Lemma 4.2 with $\delta$ replaced by $\delta_0$, to

$$g := \sum_{K \in P_{\delta_0}} f_K^{(H, \alpha, \beta)} = \sum_{J \in P_{\nu}^{(H, \alpha, \beta)}} \sum_{K \in P_{\delta_0}(J)} f_K^{(H, \alpha)}, \quad (5.9)$$

where

$$P_{\nu}^{(H, \alpha, \beta)} = \{J \in P_{\nu} : \#\{K'' \in P_{\delta_0}(J) : f_{K''}^{(H, \alpha)} \neq 0\} \in (\beta/2, \beta]\}$$

and $\kappa := q^{\lfloor \log_q \delta_0^\varepsilon \rfloor} \le \delta_0^\varepsilon$. Note that this implies

$$g_J = 1_{P_{\nu}^{(H, \alpha, \beta)}}(J) \sum_{K \in P_{\delta_0}(J)} f_K^{(H, \alpha)} \quad (5.10)$$

and $g_K = f_K^{(H, \alpha)}$ if $K$'s parent $J$ is contained in $P_{\nu}^{(H, \alpha, \beta)}$ and 0 otherwise. Write $N$ for the number of $J \in P_{\nu}$ for which $g_J \neq 0$ as in Lemma 4.2, and so $N = \#P_{\nu}^{(H, \alpha, \beta)}$. Note that by assumption the number of nonzero terms in the $\sum_K$ in (5.10) is $\sim \beta$.
With this, we then first compute

$$ \max_{K \in P_{\delta_0}} \|g_K\|_{L^{\infty}(\mathbb{Q}_q^k)}^k \sim H^k \quad (5.11) $$

as $g_K = f_K^{(H,\alpha,\beta)}$ which has height $\sim H$. Next, we have

$$ \left( \sum_{K \in P_{\delta_0}} \|g_K\|_{L^{\infty}(\mathbb{Q}_q^k)} \right)^k = \left( \sum_{J \in P_\nu} \sum_{K \in P_{\delta_0}(J)} \|g_K\|_{L^{\infty}(\mathbb{Q}_q^k)} \right)^k \quad (5.12) \\ = \left( \sum_{J \in P_\nu^{(H,\alpha,\beta)}} \sum_{K \in P_{\delta_0}(J)} \|g_K\|_{L^{\infty}(\mathbb{Q}_q^k)} \right)^k \sim (N\beta H)^k $$

as there are $N$ such $J$ for which $g_J \neq 0$ and by how $g$ is defined, each of these $J$'s that contribute has $\sim \beta$ children $K$ such that $g_K = f_K^{(H,\alpha)} \neq 0$. We can finish this estimate once again by using that $g_K$ has height $\sim H$. Third,

$$ \max_{J \in P_\nu} \left( \sum_{K \in P_{\delta_0}(J)} \|g_K\|_{L^{p-2k}(\mathbb{Q}_q^k)}^2 \right)^{(p-2k)/2} = \max_{J \in P_\nu} \left( \sum_{K \in P_{\delta_0}(J)} \|g_K\|_{L^{p-2k}(B_{\delta_0}^{-k})}^2 \right)^{(p-2k)/2} \quad (5.13) \\ \sim p,k \beta^{(p-2k)/2} H^{p-2k} \alpha \delta_0^{-k(k+1)/2} $$

as by how $g$ is defined, the $\sum_{K \in P_{\delta_0}(J)}$ has $\sim \beta$ terms and each term is made up of $\sim \alpha$ wavepackets of height $\sim H$. Note here we made use that each $T \in \mathbb{T}(K)$ has volume $\delta_0^{-k(k+1)/2}$ and $g_K$ is supported on $B_{\delta_0}^{-k}$. Finally, a similar computation gives that

$$ \left( \sum_{K \in P_{\delta_0}} \|g_K\|_{L^p(\mathbb{Q}_q^k)}^2 \right)^{p/2} = \left( \sum_{J \in P_\nu} \sum_{K \in P_{\delta_0}(J)} \|g_K\|_{L^p(B_{\delta_0}^{-k})}^2 \right)^{p/2} \sim_{p,k} (N\beta)^{p/2} H^p \alpha \delta_0^{-k(k+1)/2}. \quad (5.14) $$

Combining (5.11)-(5.14) gives that

$$ \max_{K \in P_{\delta_0}} \|g_K\|_{L^{\infty}(\mathbb{Q}_q^k)}^k \left( \sum_{K \in P_{\delta_0}} \|g_K\|_{L^{\infty}(\mathbb{Q}_q^k)} \right)^k \max_{J \in P_\nu} \left( \sum_{K \in P_{\delta_0}(J)} \|g_K\|_{L^{p-2k}(\mathbb{Q}_q^k)}^2 \right)^{(p-2k)/2} \\ \sim_{p,k} N^{-(p-2k)/2} \left( \sum_{K \in P_{\delta_0}} \|g_K\|_{L^p(\mathbb{Q}_q^k)}^2 \right)^{p/2} . $$

Using this with Lemma 4.2 where $g$ is as given in (5.9), then shows that
$$ \int_{Q_q^k} |g|^p \le C \mathfrak{D}_p (\frac{\delta_0}{\kappa})^p \left( \sum_{K \in P_{\delta_0}} \|g_K\|_{LP(Q_q^k)}^2 \right)^{p/2} \\ + C q^{-k(k-1)} \kappa^{-(k^2+4k-2)} \nu^{-\frac{k(k-1)}{2}} N^{\frac{p-2k}{2}} \mathfrak{D}_{p-2k} \left( \frac{\delta_0}{\nu} \right)^{p-2k} \left( \sum_{K \in P_{\delta_0}} \|g_K\|_{LP(Q_q^k)}^2 \right)^{p/2} . $$

Note $\frac{\delta_0}{\kappa} \ge \frac{\delta_0}{\delta_0^\epsilon} = \delta_0^{1-\epsilon} \ge \delta^{1-\epsilon}$, so $\mathfrak{D}_p(\frac{\delta_0}{\kappa}) \le D_p(\frac{\delta_0}{\kappa}) \le D_p(\delta^{1-\epsilon})$ where in the second inequality, we have used monotonicity. Similarly, $\frac{\delta_0}{\nu} \ge \frac{\delta_0}{\delta_0^{1/k}} = \delta_0^{1-\frac{1}{k}} \ge \delta^{1-\frac{1}{k}}$, so $\mathfrak{D}_{p-2k}(\frac{\delta_0}{\nu}) \le D_{p-2k}(\delta^{1-\frac{1}{k}})$. As a result,

$$ \int_{Q_q^k} |g|^p \le CD_p(\delta^{1-\epsilon})^p \left( \sum_{K \in P_{\delta_0}} \|g_K\|_{LP(Q_q^k)}^2 \right)^{p/2} \\ + C q^{-k(k-1)} \kappa^{-(k^2+4k-2)} \nu^{-\frac{k(k-1)}{2}} N^{\frac{p-2k}{2}} D_{p-2k}(\delta^{1-\frac{1}{k}})^{p-2k} \left( \sum_{K \in P_{\delta_0}} \|g_K\|_{LP(Q_q^k)}^2 \right)^{p/2} . $$

Now use $N \le \nu^{-1}$ and $\|g_K\|_{LP(Q_q^k)} = \|f_K^{(H,\alpha,\beta)}\|_{LP(Q_q^k)} \le \|f_K\|_{LP(Q_q^k)}$. Thus,

$$ \int_{Q_q^k} |f|^p \le C(\log \delta^{-1})^{3p} \times \\ \left[ D_p(\delta^{1-\epsilon})^p + q^{-k(k-1)} \kappa^{-(k^2+4k-2)} \nu^{-\left(\frac{p}{2} + \frac{k(k-3)}{2}\right)} D_{p-2k}(\delta^{1-\frac{1}{k}})^{p-2k} \right] \left( \sum_{K \in P_{\delta_0}} \|f_K\|_{LP(Q_q^k)}^2 \right)^{p/2} . $$

But $\nu^{-1} \le q\delta_0^{-1/k} \le q\delta^{-1/k}$ and $\kappa^{-1} \le q\delta_0^{-\epsilon} \le q\delta^{-\epsilon}$. This completes the proof of (5.2). $\square$

<figure><img src="image_2.png" /></figure>

## 5.2 | Proof of Theorem 1.1

We now finish the proof of Theorem 1.1. It suffices to iterate (5.2) by using an induction on $p$ and induction on $\delta$. Applying the definition of $D_p(\delta)$ from (5.1) and the hypothesis of Theorem 1.1 gives that

$$ D_{p_0}(\delta)^{p_0} \le C_1 \delta^{-(\frac{p_0}{2} - \frac{k(k+1)}{2})} - c(p_0)(1 - \frac{1}{k})^{p_0/(2k)} $$

for all $\delta \in (0, 1)$ and some $c(p_0) \ge 0$ such that the power of $\delta^{-1}$ is nonnegative. Note that from (1.9), $a(p, p_0) = a(p - 2k, p_0) + \frac{p}{2} + \frac{k^2+7k-4}{2}$ and $a(p_0, p_0) = 0$. Additionally, (1.10) gives $b(p_0) \ge 0$ where

$$ b(p) := (p - k(k + 1))(1 - \frac{1}{k})^{-\frac{p}{2k}} + 2c(p_0), $$
and $b(p)$ is an increasing function of $p$ on $[2, \infty)$: indeed, for $p \ge 2$, we have

$$
\begin{align*}
b'(p) &= \left(1 - \frac{1}{k}\right)^{-\frac{p}{2k}} \left[1 + \frac{p - k(k + 1)}{2k} \log\left(1 - \frac{1}{k}\right)\right]^{-1} \\
&= \left(1 - \frac{1}{k}\right)^{-\frac{p}{2k}} \left[1 + \frac{p - k(k + 1)}{2k} \left(\log k - \log(k - 1)\right)\right] \\
&\ge \left(1 - \frac{1}{k}\right)^{-\frac{p}{2k}} \left[1 + \frac{2 - k(k + 1)}{2k} \left(\log k - \log(k - 1)\right)\right] \\
&\ge \left(1 - \frac{1}{k}\right)^{-\frac{p}{2k}} \left[1 + \frac{2 - k(k + 1)}{2k} \frac{1}{k - 1}\right],
\end{align*}
$$

where we used $p \ge 2$ in the first inequality, and used $\log k - \log(k-1) \le \frac{1}{k-1}$ with $2 - k(k+1) \le 0$ in the second inequality. This gives

$$
b'(p) \geq \left(1 - \frac{1}{k}\right)^{-\frac{p}{2k}} \left[1 + \frac{2 - k - k^2}{2k(k - 1)}\right] = \left(1 - \frac{1}{k}\right)^{-\frac{p}{2k}} \left[1 - \frac{k + 2}{2k}\right] \geq 0
$$

since $k \ge 2$, proving that $b(p)$ is an increasing function of $p$ on $[2, \infty)$. As a result, from $b(p_0) \ge 0$, we see that $b(p) \ge 0$ for all $p \ge p_0$, and hence

$$
\frac{p}{2} - \frac{k(k+1)}{2} + c(p_0)(1 - \frac{1}{k})^{\frac{p}{2k}} \ge 0 \quad (5.15)
$$

for all $p \ge p_0$.

Assume for every $0 < \varepsilon < 1$ and all $\delta \in (0, 1)$ we know

$$
D_{p-2k}(\delta)^{p-2k} \le C_{p-2k,\varepsilon} q^{a(p-2k,p_0)} \delta^{-\left(\frac{p-2k}{2} - \frac{k(k+1)}{2}\right)} - c(p_0)\left(1-\frac{1}{k}\right)^{\frac{p-2k}{2k}} - \varepsilon
$$

for some $p \in p_0 + 2k \mathbb{N}$ (this is true for $p = p_0 + 2k$) and $C_{p-2k,\varepsilon}$ is allowed to depend on $C_1$. Then (5.2) gives

$$
\begin{align*}
D_p(\delta)^p &\le C(\log \delta^{-1})^{3p} [D_p(\delta^{1-\varepsilon})^p \\
&\qquad + C_{p-2k,\varepsilon} q^{a(p,p_0)} \delta^{-(k^2+4k-2)\varepsilon} \delta^{-\frac{1}{k}\left(\frac{p}{2} + \frac{k(k-3)}{2}\right)} \delta^{-(1-\frac{1}{k})\left(\frac{p-2k}{2} - \frac{k(k+1)}{2}\right)} - c(p_0)(1-\frac{1}{k})^{\frac{p-2k}{2k}+1-\varepsilon}] \\
&= C(\log \delta^{-1})^{3p} D_p(\delta^{1-\varepsilon})^p + C C_{p-2k,\varepsilon} q^{a(p,p_0)} \delta^{-\left(\frac{p}{2} - \frac{k(k+1)}{2}\right)} - c(p_0)(1-\frac{1}{k})^{\frac{p}{2k}} - (k^2+4k)\varepsilon
\end{align*}
$$

for all $\delta, \varepsilon \in (0, 1)$ where $C$ here depends only on $k$ and $p$. Iterating this inequality $M$ times with $M$ to be chosen later gives that

$$
D_p(\delta)^p \le C^M (\log \delta^{-1})^{3Mp} D_p(\delta^{(1-\varepsilon)^M})^p \\
+ CC_{p-2k,\varepsilon} q^{a(p,p_0)} \delta^{-(k^2+4k)\varepsilon} \sum_{j=0}^{M-1} C^j (\log \delta^{-1})^{3pj} \delta^{-(1-\varepsilon)^j} \left[ \left(\frac{p}{2} - \frac{k(k+1)}{2}\right) + c(p_0) \left(1 - \frac{1}{k}\right)^{\frac{p}{2k}} \right].
$$
Trivially, we have $D_p(\delta^{(1-\varepsilon)^M}) \le \delta^{-(1-\varepsilon)^M/2}$. Thus,

$$D_p(\delta)^p \le C^M (\log \delta^{-1})^{3M} p \delta^{-(1-\varepsilon)^M p/2} + CC_{p-2k, \varepsilon} q^{a(p, p_0)} \delta^{-(k^2+4k)\varepsilon} \sum_{j=0}^{M-1} C^j (\log \delta^{-1})^{3pj} \delta^{-(1-\varepsilon)^j} \left[ \left( \frac{p}{2} - \frac{k(k+1)}{2} \right) + c(p_0) \left( 1 - \frac{1}{k} \right)^{\frac{p}{2k}} \right]. \quad (5.16)$$

By (5.15), the power of $\delta^{-1}$ in (5.16) is positive and so using that $(1 - \varepsilon)^j \le 1$, the sum can be controlled by

$$MC^M (\log \delta^{-1})^{3M} p \delta^{-(\frac{p}{2} - \frac{k(k+1)}{2}) - c(p_0)\left(1 - \frac{1}{k}\right)^{\frac{p}{2k}}}.$$

Inserting this into (5.16) and choosing $M$ be the least integer such that $(1 - \varepsilon)^M \le \varepsilon$ (and so $M = \lceil \frac{\log \varepsilon^{-1}}{\log(1-\varepsilon)^{-1}} \rceil$) then shows that

$$D_p(\delta) \lesssim_{p,\varepsilon,C_1} q^{a(p,p_0)/p} (\log \delta^{-1})^{3M} \delta^{-(\frac{1}{2} - \frac{k(k+1)}{2p}) - \frac{c(p_0)}{p} \left(1 - \frac{1}{k}\right)^{\frac{p}{2k}} - \frac{(k^2+4k)\varepsilon}{p}}$$

for all $\delta, \varepsilon \in (0, 1)$. As $(\log \delta^{-1})^{3M} \lesssim_{\varepsilon} \delta^{-\varepsilon}$, by redefining $\varepsilon$ we have

$$D_p(\delta) \lesssim_{p,\varepsilon,C_1} q^{a(p,p_0)/p} \delta^{-(\frac{1}{2} - \frac{k(k+1)}{2p}) - \frac{c(p_0)}{p} \left(1 - \frac{1}{k}\right)^{\frac{p}{2k}} - \varepsilon}.$$

## ACKNOWLEDGMENTS

This question was first posed to the third and sixth author by Shaoming Guo when the third author was visiting the Department of Mathematics at the Chinese University of Hong Kong in July 2019. This question was posed again by Shaoming Guo during a problem session at the Arithmetic (and) Harmonic Analysis workshop held (virtually) at the Mittag-Leffler Institute in early June 2021 and this current collaboration arose from that particular workshop.

Kevin Hughes is supported by the Additional Funding Programme for Mathematical Sciences, delivered by EPSRC (EP/V521917/1) and the Heilbronn Institute for Mathematical Research. Zane Kun Li is supported by NSF Grant DMS-1902763. Akshat Mudgal is supported by Ben Green's Simons Investigator Grant, ID 376201; Olivier Robert is supported by the joint FWF-ANR Project Arithmand: FWF: I 4945-N and ANR-20-CE91-0006; and Po-Lam Yung is supported by a Future Fellowship FT20010039 from the Australian Research Council. Zane Kun Li would also like to thank the National Center for Theoretical Sciences (NCTS) in Taipei, Taiwan for their kind hospitality during his visit, where part of this work was written. The authors also acknowledge kind support from the American Institute of Mathematics through the Fourier restriction research community.

## JOURNAL INFORMATION

*Mathematika* is owned by University College London and published by the London Mathematical Society. All surplus income from the publication of *Mathematika* is returned to mathematicians and mathematics research via the Society's research grants, conference grants, prizes, initiatives for early career researchers and the promotion of mathematics.
ORCID

Kevin Hughes https://orcid.org/0000-0002-8621-8259

Zane Kun Li https://orcid.org/0000-0003-0202-0884

Akshat Mudgal https://orcid.org/0000-0001-6043-6576

Po-Lam Yung https://orcid.org/0000-0002-0441-3625

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## APPENDIX: PROOF OF $\mathfrak{D}_{2k}(\delta) \lesssim_{\varepsilon} \delta^{-\varepsilon}$

Fix $k \in \mathbb{N}$ and a prime $q > k$. For $\delta \in q^{-\mathbb{N}}$, let $S(\delta)$ be the smallest constant such that the reverse square function estimate

$$ \int_{\mathbb{Q}_q^k} |g|^{2k} \le S(\delta)^{2k} \int_{\mathbb{Q}_q^k} \left( \sum_{K \in P_\delta} |g_K|^2 \right)^k $$

holds for every Schwartz function $g$ on $\mathbb{Q}_q^k$ with Fourier transform supported in $\bigcup_{K \in P_\delta} \theta_K$. We will prove that

$$ S(\delta) \lesssim_{\varepsilon} \delta^{-\varepsilon} $$

for every $\varepsilon > 0$, which by Minkowski's inequality is stronger than the assertion $\mathfrak{D}_{2k}(\delta) \lesssim_{\varepsilon} \delta^{-\varepsilon}$.

Let $\delta \in q^{-\mathbb{N}}$, $g$ be as above, and $\kappa \in q^{-\mathbb{N}} \cap [\delta, 1]$. The broad/narrow dichotomy given by the pointwise estimate (4.2) implies

$$ \int_{\mathbb{Q}_q^k} |g|^{2k} \le 2^{2k-1} k^{2k} \sum_{I \in P_\kappa} \int_{\mathbb{Q}_q^k} |g_I|^{2k} + 2^{2k-1} \kappa^{-(4k-2)} \sum_{\substack{I_1, \dots, I_k \in P_\kappa \\ d(I_i, I_j) > \kappa \forall i \ne j}} \int_{\mathbb{Q}_q^k} |g_{I_1} \cdots g_{I_k}|^2 \quad (A.1) $$

Furthermore, by a rescaling argument similar to that in Lemma 4.1, we have

$$ \sum_{I \in P_\kappa} \int_{\mathbb{Q}_q^k} |g_I|^2 \le S\left(\frac{\delta}{\kappa}\right)^{2k} \sum_{I \in P_\kappa} \int_{\mathbb{Q}_q^k} \left( \sum_{K \in P_\delta(I)} |g_K|^2 \right)^k \le S\left(\frac{\delta}{\kappa}\right)^{2k} \int_{\mathbb{Q}_q^k} \left( \sum_{K \in P_\delta} |g_K|^2 \right)^k \quad (A.2) $$
where we used the pointwise inequality $\sum_{I \in P_\kappa} (\sum_{K \in P_\delta(I)} |g_K|^2)^k \le (\sum_{K \in P_\delta} |g_K|^2)^k$ in the last inequality. To proceed further, fix now $I_1, \dots, I_k \in P_\kappa$ with $d(I_i, I_j) > \kappa$ for all $i \ne j$. We expand

$$ \int_{\mathbb{Q}_q^k} |g_{I_1} \dots g_{I_k}|^2 = \sum_{K_i \in P_\delta(I_i)} \sum_{\bar{K}_j \in P_\delta(I_j)} \int_{\mathbb{Q}_q^k} g_{K_1} \dots g_{K_k} \overline{g_{\bar{K}_1}} \dots \overline{g_{\bar{K}_k}} $$

and write

$$ \int_{\mathbb{Q}_q^k} g_{K_1} \dots g_{K_k} \overline{g_{\bar{K}_1}} \dots \overline{g_{\bar{K}_k}} = [\widehat{g_{K_1}} * \dots * \widehat{g_{K_k}} * \widehat{g_{\bar{K}_1}} * \dots * \widehat{g_{\bar{K}_k}}](0). $$

For each $\bar{K}_1 \in P_\delta(I_1), \dots, \bar{K}_k \in P_\delta(I_k)$, we count the number of ordered $k$-tuples $(K_1, \dots, K_k)$ with $K_i \in P_\delta(I_i)$ for $i=1, \dots, k$ and $0 \in \text{supp}(\widehat{g_{\bar{K}_1}} * \dots * \widehat{g_{\bar{K}_k}} * \widehat{g_{\bar{K}_1}} * \dots * \widehat{g_{\bar{K}_k}})$. The proof of Lemma 4.4 shows that the number of such ordered $k$-tuples is $\le (q\kappa)^{-k(k-1)}$ (in fact, here we only need that $\widehat{g_{\bar{K}_j}}$ is supported in the cube $\tau_{K_j}$ rather than the smaller parallelepiped $\theta_{K_j}$). So, using Cauchy-Schwarz,

$$ \sum_{K_i \in P_\delta(I_i)} \sum_{\bar{K}_j \in P_\delta(I_j)} \int_{\mathbb{Q}_q^k} g_{K_1} \dots g_{K_k} \overline{g_{\bar{K}_1}} \dots \overline{g_{\bar{K}_k}} \le (q\kappa)^{-k(k-1)} \sum_{K_i \in P_\delta(I_i)} \int_{\mathbb{Q}_q^k} |g_{K_1} \dots g_{K_k}|^2. $$

It follows that

$$ \sum_{\substack{I_1, \dots, I_k \in P_\kappa \\ d(I_i, I_j) > \kappa \forall i \ne j}} \int_{\mathbb{Q}_q^k} |g_{I_1} \dots g_{I_k}|^2 \le (q\kappa)^{-k(k-1)} \int_{\mathbb{Q}_q^k} \left( \sum_{K \in P_\delta} |g_K|^2 \right)^k. \quad (A.3) $$

Alternatively, multilinear restriction estimate and $L^2$ orthogonality says that for any ball $B_{\delta^{-1}}$ of radius $\delta^{-1}$ in $\mathbb{Q}_q^k$, one has

$$ \int_{B_{\delta^{-1}}} |g_{I_1} \dots g_{I_k}|^2 \lesssim_{\kappa} (\delta^{k-1})^k \prod_{j=1}^k \int_{B_{\delta^{-1}}} |g_{I_j}|^2 = |B_{\delta^{-1}}|^{-(k-1)} \prod_{j=1}^k \int_{B_{\delta^{-1}}} \left( \sum_{K_j \in P_\delta(I_j)} |g_{K_j}|^2 \right), $$

and as each $|g_{K_j}|$ is constant on $B_{\delta^{-1}}$, we have

$$ |B_{\delta^{-1}}|^{-(k-1)} \prod_{j=1}^k \int_{B_{\delta^{-1}}} \left( \sum_{K_j \in P_\delta(I_j)} |g_{K_j}|^2 \right) = \int_{B_{\delta^{-1}}} \prod_{j=1}^k \left( \sum_{K_j \in P_\delta(I_j)} |g_{K_j}|^2 \right). $$

Summing over all $B_{\delta^{-1}} \subset \mathbb{Q}_q^k$ and all $I_1, \dots, I_k \in P_\kappa$, we have

$$ \sum_{\substack{I_1, \dots, I_k \in P_\kappa \\ d(I_i, I_j) > \kappa \forall i \ne j}} \int_{\mathbb{Q}_q^k} |g_{I_1} \dots g_{I_k}|^2 \lesssim_{\kappa} \int_{\mathbb{Q}_q^k} \left( \sum_{K \in P_\delta} |g_K|^2 \right)^k, $$

which for the purposes below is as good as (A.3). Putting (A.2) and (A.3) back into (A.1), we have

$$ S(\delta)^{2k} \le 2^{2k-1} k^{2k} S(\frac{\delta}{\kappa})^{2k} + 2^{2k-1} \kappa^{-(4k-2)} (q\kappa)^{-k(k-1)}. $$
Iterating this gives

$$S(\delta)^{2k} \le (2^{2k-1} k^{2k})^N S(\frac{\delta}{\kappa^N})^{2k} + N 2^{2k-1} \kappa^{-(4k-2)} (q\kappa)^{-k(k-1)}$$

for all positive integers N for which $\kappa^N \ge \delta$; in particular, applying this with $N = \lfloor \frac{\log \delta^{-1}}{\log \kappa^{-1}} \rfloor$, and noting that $S(\delta/\kappa^N) \le (\delta/\kappa^N)^{-1/2} \le \kappa^{-1/2}$, we have

$$S(\delta)^{2k} \le \delta^{-\frac{\log(2^{2k-1}k^{2k})}{\log \kappa^{-1}}} \kappa^{-k} + \frac{\log \delta^{-1}}{\log \kappa^{-1}} 2^{2k-1} \kappa^{-(4k-2)} (q\kappa)^{-k(k-1)}.$$

By choosing $\kappa = \kappa(\varepsilon)$ sufficiently small so that $\frac{\log(2^{2k-1}k^{2k})}{\log \kappa^{-1}} \le 2k\varepsilon$, one obtains $S(\delta) \lesssim \delta^{-\varepsilon}$, as desired.