# A non-archimedean variant of Littlewood–Paley theory for curves

Jonathan Hickman  School of Mathematics and Maxwell Institute for Mathematical
Sciences, University of Edinburgh, James Clerk Maxwell Building, Kings
Buildings, Peter Guthrie Tait Road, Edinburgh, EH9 3FD, UK.
<EMAIL_ADDRESS>and  James Wright  School of Mathematics and Maxwell
Institute for Mathematical Sciences, University of Edinburgh, James Clerk
Maxwell Building, Kings Buildings, Peter Guthrie Tait Road, Edinburgh, EH9
3FD, UK<EMAIL_ADDRESS>
###### Abstract.

We prove a variant of a square function estimate for the extension operator
associated to the moment curve in non-archimedean local fields. The arguments
rely on a structural analysis of congruences (sublevel sets) of univariate
polynomials over field extensions of the base field. Our analysis can be
adapted to the archimedean setting as well.

##  1\. Introduction

###  1.1. Statement of the results

This paper concerns the Fourier restriction theory for curves and associated
Littlewood–Paley-type inequalities. Classically, this theory forms part of
Euclidean harmonic analysis, however here we explore these questions in the
setting of a general locally compact topological field $K$ with a nontrivial
topology. Such fields carry a natural absolute value $|\,\cdot\,|_{K}$ and a
Haar measure $\mu$. They are classified as archimedean local fields (when
$K={\mathbb{R}}$ is the real field or when $K={\mathbb{C}}$ is the complex
field) or non-archimedean local fields such as the $p$-adic field
${\mathbb{Q}}_{p}$. The Littlewood–Paley theory for curves is well known when
$K={\mathbb{R}}$ is the real field so we will state and prove our results for
non-archimedean local fields. In an appendix we will show how to adapt our
arguments to work in the archimedean setting.

Let $(K,|\,\cdot\,|_{K})$ be a non-archimedean local field with ring of
integers $\mathfrak{o}_{K}$, residue class field $k_{K}$, uniformiser
$\pi_{K}$ and $q_{K}:=|\pi|_{K}^{-1}$. For the reader’s convenience, we will
review some of the basic concepts of analysis over local fields in §2 below.
Fix an additive character $e\colon K\to\mathbb{C}$ such that $e$ restricts to
the constant function 1 on $\mathfrak{o}_{K}$ and to a non-principal character
on $\pi_{K}^{-1}\mathfrak{o}_{K}$. For $n\geq 2$, we define the extension
operator associated to the moment curve by

$Ef(\mathbf{x}):=\int_{\mathfrak{o}_{K}}e(x_{1}t+x_{2}t^{2}+\cdots+x_{n}t^{n})f(t)\,\mathrm{d}\mu(t)\qquad\textrm{for
all $f\in L^{1}(K)$ and $\mathbf{x}\in K^{n}$.}$ (1)

Here and below, integration is taken with respect to the Haar measure $\mu$ on
$K$, which is normalised so that $\mu(\mathfrak{o}_{K})=1$.

The operator $E$ is a fundamental object of study in the Fourier restriction
theory over local fields $K$. This theory was investigated systematically by
the authors in [7], with a focus on the problem of determining Lebesgue space
mapping properties. Here we are interested in Littlewood–Paley or square
function inequalities for the operator (1). To describe the setup, fix
$\alpha\in\mathbb{N}$ and let $\mathcal{I}(q_{K}^{-\alpha})$ denote the
collection of $q_{K}^{\alpha}$ distinct balls of the form

$B_{K}(x;q_{K}^{-\alpha}):=\\{t\in\mathfrak{o}_{K}:|t-x|_{K}\leq
q_{K}^{-\alpha}\\},\qquad x\in\mathfrak{o}_{K}.$

Thus, $\mathcal{I}(q_{K}^{-\alpha})$ defines a decomposition of
$\mathfrak{o}_{K}$, which induces a decomposition of the extension operator

$Ef=\sum_{I\in\mathcal{I}(q_{K}^{-\alpha})}E_{I}f\qquad\textrm{where
$E_{I}f:=E\big{(}f\chi_{I}\big{)}$ for all
$I\in\mathcal{I}(q_{K}^{-\alpha})$.}$ (2)

Here $\chi_{I}$ denotes the characteristic function of
$I\in\mathcal{I}(q_{K}^{-\alpha})$.

######  Theorem 1.1.

Let $(K,|\,\cdot\,|_{K})$ be a non-archimedean local field and
$\mathrm{char}\,k_{K}>n\geq 2$. For all $1\leq m\leq n$ and all
$\alpha\in\mathbb{N}$, the identity

$\|Ef\|_{L^{2m}(B(\mathbf{x},q_{K}^{\alpha
n}))}\leq(m!)^{1/2m}\Big{\|}\big{(}\sum_{I\in\mathcal{I}(q_{K}^{-\alpha})}|Ef_{I}|^{2}\big{)}^{1/2}\Big{\|}_{L^{2m}(B(\mathbf{x},q_{K}^{\alpha
n}))}$

holds for all $f\in L^{1}(\mathfrak{o}_{K})$ and all $\mathbf{x}\in K^{n}$.

Throughout the paper, $L^{p}$ norms are taken with respect to the Haar measure
on $K^{n}$ given by the $n$-fold product of $\mu$ above. The balls
$B(\mathbf{x},q_{K}^{\alpha n})$ are defined with respect to the
$\ell^{\infty}$ norm induced by $|\,\cdot\,|_{K}$: see §2 for further details.

Theorem 1.1 is an analogue of a Euclidean result from [12, 11, 6], as
described below in §1.2. Moreover, recently square function inequalities of
this type were investigated in the local field setting in [1] in the case
$n=2$ for general polynomial curves.111The methods of [1] imply bounds for
$n\geq 2$ but only at the level of $m=2$.

By a well-known $2n$-orthogonality argument due to Córdoba and Fefferman, the
proof of Theorem 1.1 reduces to establishing the following number-theoretic
proposition.

######  Proposition 1.2.

Let $(K,|\,\cdot\,|_{K})$ be a non-archimedean local field,
$\mathrm{char}\,k_{K}>n\geq 2$ and $a\in\mathbb{N}$. Suppose
$(x_{1},\dots,x_{n})$, $(y_{1},\dots,y_{n})\in(\mathfrak{o}_{K})^{n}$ satisfy

$|x_{1}^{j}+\cdots+x_{n}^{j}-y_{1}^{j}-\cdots-y_{n}^{j}|_{K}\leq
q_{K}^{-na}\qquad\textrm{for $1\leq j\leq n$.}$ (3)

Then there exists a permutation $\sigma$ on $\\{1,\cdots,n\\}$ such that
$|x_{j}-y_{\sigma(j)}|_{K}\leq q_{K}^{-a}$ for all $1\leq j\leq n$.

Proposition 1.2 examines the structure of ‘almost solutions’ to a Vinogradov-
type system of equations. In particular, it can be roughly interpreted as
saying that every ‘almost solution’ to the system
$x_{1}^{j}+\cdots+x_{n}^{j}=y_{1}^{j}+\cdots+y_{n}^{j}$ for $1\leq j\leq n$ is
‘almost trivial’. A similar statement appears in the Euclidean context in [6],
although the method of proof used in [6] breaks down completely in the non-
archimedean setting (see the discussion in §1.3 below).

###  1.2. Motivation: the Euclidean case

It is instructive to contrast Theorem 1.1 with counterpart results in the
Euclidean setting. For $n\geq 2$ let $\gamma\colon[0,1]\to\mathbb{R}^{n}$ be a
$C^{n}$ curve in $n$-dimensional Euclidean space which satisfies the non-
degeneracy hypothesis $\det(\gamma^{\prime}(t)\cdots\gamma^{(n)}(t))\neq 0$
for all $t\in[0,1]$. Define the associated extension operator

$Ef(\mathbf{x}):=\int_{0}^{1}e^{2\pi
i\mathbf{x}\cdot\gamma(t)}f(t)\,\mathrm{d}t\qquad\textrm{for all $f\in
L^{1}([0,1])$ and $\mathbf{x}\in\mathbb{R}^{n}$.}$

Let $0<\delta\leq 1$ be a dyadic number and $\mathcal{I}(\delta)$ be the
decomposition of $[0,1]$ into closed dyadic intervals of length $\delta$. We
decompose the extension operator as in (2), with $q_{K}^{-\alpha}$ replaced by
$\delta$. Under these hypotheses, it is known that for each $1\leq m\leq n$
there exists a constant $C_{m}\geq 1$ such that

$\|Ef\|_{L^{2m}(B_{\delta^{-n}})}\leq
C_{m}\Big{\|}\big{(}\sum_{I\in\mathcal{I}(\delta)}|Ef_{I}|^{2}\big{)}^{1/2}\Big{\|}_{L^{2m}(w_{B_{\delta^{-n}}})}$
(4)

holds for all $f\in L^{1}([0,1])$. Here $B_{\delta^{-n}}$ is a Euclidean ball
of radius $\delta^{-n}$ and arbitrary centre, and $w_{B_{\delta^{-n}}}$ is a
rapidly decaying weight function concentrated on $B_{\delta^{-n}}$; we refer
to [6] for the precise definitions. The inequality in the $n=2$ case goes back
to work of Fefferman [5]. The general case is implicit in works of Prestini
[12, 11], albeit the arguments of these papers are somewhat lacking in detail.
More recently, the inequality was rediscovered in [6], which includes a
complete proof and contextualises the result in relation to recent
developments in harmonic analysis and analytic number theory. It is remarked
that a reverse form of (4) holds as a simple consequence of a classical and
elementary square function estimate due to Carleson (see, for instance, [13]).

Interest in bounds such as (4) has been spurred by the breakthrough work of
Bourgain–Demeter–Guth [2] which settled the long-standing main conjecture in
Vinogradov’s mean value theorem, a central problem in the theory of
Diophantine equations. The approach in [2] relied on establishing certain
decoupling estimates for the extension operator associated to the moment
curve. These estimates are of a superficially similar form to the inequality
in (4), although (4) is much more elementary than the key estimate from [2]
and is not sufficient to prove the main conjecture. Nevertheless, in [6] the
authors discuss a general philosophy relating square function bounds to
Diophantine equations.

###  1.3. Remarks on the proof

Recall that the key ingredient in the proof of Theorem 1.1 is Proposition 1.2.
The latter is a natural non-archimedean analogue of Proposition 1.3 from [6].
It is remarked that the arguments used in [6] rely heavily on the order
structure of the real line and break down completely in the non-archimedean
setting. Consequently, to establish Proposition 1.2 we use a markedly
different approach which is more algebraic in nature and involves the
geometric analysis of sublevel sets, corresponding to a structural analysis of
polynomial congruences.

To describe the rudiments of our approach, we first consider the following
reformulation of Proposition 1.2 in the case where $K=\mathbb{Q}_{p}$ is the
field of $p$-adic numbers.

######  Corollary 1.3.

Let $n$, $a\in\mathbb{N}$ and $p$ be a rational prime such that $p>n\geq 2$.
Suppose $(x_{1},\dots,x_{n})$, $(y_{1},\dots,y_{n})\in\mathbb{Z}^{n}$ satisfy
the congruence equations

$x_{1}^{j}+\cdots+x_{n}^{j}\equiv y_{1}^{j}+\cdots+y_{n}^{j}\mod
p^{na}\qquad\textrm{for $1\leq j\leq n$.}$ (5)

Then there exists a permutation $\sigma$ on $\\{1,\cdots,n\\}$ such that
$x_{j}\equiv y_{\sigma(j)}\mod p^{a}$ for all $1\leq j\leq n$.

Corollary 1.3 is easily verified for $a=1$.222Moreover, when $a=1$ one need
only assume (5) holds with $p^{n}$ replaced with $p$. Indeed, this case holds
as a consequence of the classical Girard–Newton formulæ together with
uniqueness of factorisation of polynomials over the field
$\mathbb{Z}/p\mathbb{Z}$: see, for example, [9]. To prove the general case of
Corollary 1.3, we will still make use of the Girard–Newton formulæ. However,
we must now consider polynomials over the rings $\mathbb{Z}/p^{a}\mathbb{Z}$
for $a\geq 2$ and therefore cannot rely on uniqueness of factorisation.

The key tool used to overcome these issues is a refined version of the
Phong–Stein–Sturm sublevel set decomposition [10], formulated over non-
archimedean local fields. It can be viewed as a refined structural description
of polynomial congruences, extending work of Chalk [4] which is valid for
large values of $a$ and work of Stewart [14] for polynomials with a nonzero
discriminant. This decomposition has been applied previously by the second
author to study complete exponential sums and congruence equations [16, 17]
and is recalled in Lemma 3.1 below. A slightly curious feature of the argument
is that we apply the sublevel set decomposition over a high degree field
extension of $K$ rather than $K$ itself.

###  1.4. Archimedean fields

Our arguments can be adapted to work in the archimedean setting. As a
consequence, we obtain a new proof of the Euclidean estimate (4) for the
moment curve. Moreover, we are also able to prove an analogue of (4) when
$K={\mathbb{C}}$ is the complex field. In this case, $E$ is the extension
operator associated to a certain $2$-surface in
${\mathbb{R}}^{2n}\simeq{\mathbb{C}}^{n}$. For $n=2$ this complex estimate is
contained in [1], but it appears to be new in higher dimensions. Adapting the
proofs to the archimedean setting is not entirely straightforward, and a
discussion of the necessary modifications is provided in Appendix A.

### Notation

Depending on the context, $|A|$ will either denote the absolute value of a
complex number $A$ or the cardinality of a finite set $A$.

##  2\. Review of the basic concepts from the theory of local fields

###  2.1. Non-archimedean local fields

A valued field  $(K,|\,\cdot\,|_{K})$ is a field $K$ together with an absolute
value map $|\,\cdot\,|_{K}\colon K\to[0,\infty)$ satisfying

  1. i)

$|x|_{K}=0$ if and only if $x=0$;

  2. ii)

$|xy|_{K}=|x|_{K}|y|_{K}$ for all $x$, $y\in K$;

  3. iii)

$|x+y|_{K}\leq|x|_{K}+|y|_{K}$ for all $x$, $y\in K$.

The absolute value $|\,\cdot\,|_{K}$ is non-archimedean if iii) can be
strengthened to

  1. iii′)

$|x+y|_{K}\leq\max\\{|x|_{K},|y|_{K}\\}$ for all $x,y\in K$,

otherwise it is archimedean. Note that any field $K$ admits a trivial absolute
value is given by $|x|_{K}=1$ for all $x\in K^{*}$ (the group of units) and
$|0|_{K}=0$.

A valued field $(K,|\,\cdot\,|_{K})$ is endowed with a metric $d_{K}$ by
setting $d_{K}(x,y):=|x-y|_{K}$ for all $x$, $y\in K$. For a non-archimedean
absolute value $d$ is an ultrametric, satisfying the ultrametric triangle
inequality $d_{K}(x,z)\leq\max\\{d_{K}(x,y),d_{K}(y,z)\\}$ for all $x$, $y$,
$z\in K$. The ball centred at $x\in K$ of radius $r>0$ is defined by

$B_{K}(x,r):=\\{y\in K:|y-x|_{K}\leq r\\}.$

Henceforth let $(K,|\,\cdot\,|_{K})$ be a valued field where $|\,\cdot\,|_{K}$
is a non-trivial, non-archimedean absolute. The ring of integers of $K$ is
defined as

$\mathfrak{o}_{K}:=\\{x\in K:|x|_{K}\leq 1\\};$

it is easy to see $\mathfrak{o}_{K}$ is a local ring with unique maximal ideal

$\mathfrak{m}_{K}:=\\{x\in K:|x|_{K}<1\\}.$

The residue class field of $K$ is defined to be the quotient
$k_{K}:=\mathfrak{o}_{K}/\mathfrak{m}_{K}$. Finally, the value group
$\Gamma_{K}:=\\{|x|_{K}\in(0,\infty):x\in K^{*}\\}$ is the multiplicative
subgroup of $(0,\infty)$ formed by the image of $K^{*}$ under
$|\,\cdot\,|_{K}$.

The absolute value $|\,\cdot\,|_{K}$ is discrete if the group $\Gamma_{K}$ is
discrete. This holds if and only if the maximal ideal $\mathfrak{m}_{K}$ is
principal. In this case, we let $\pi_{K}\in\mathfrak{m}_{K}$ denote some
choice of generator, which is referred to as a uniformiser for $K$. It follows
that $\Gamma_{K}=\\{q_{K}^{-\nu}:\nu\in\mathbb{Z}\\}$ where
$q_{K}:=|\pi_{K}|_{K}^{-1}\in(1,\infty)$.

######  Definition 2.1.

A valued field $(K,|\,\cdot\,|_{K})$ is a non-archimedean local field if
$|\,\cdot\,|_{K}$ is a non-trivial discrete non-archimedean absolute value, it
is complete and the residue class field $k_{K}$ is finite.

If $(K,|\,\cdot\,|_{K})$ is a non-archimedean local field, then
$\mathfrak{o}_{K}$ is a compact subset of $K$ and, consequently, $K$ is a
locally compact metric space. If we fix $\pi_{K}$ a uniformiser for $K$ and
$\mathcal{A}\subseteq\mathfrak{o}_{K}$ a set of representatives for $k_{K}$,
then every $x\in K^{*}$ can be written uniquely as
$x=\sum_{m=M}^{\infty}x_{m}\pi_{K}^{m}$ for some sequence
$(x_{m})_{m=M}^{\infty}$ of elements from $\mathcal{A}$. Here the series is
understood to converge with respect to the metric $d$ introduced above. It
follows that each ball $B_{K}(x,q_{K}^{-\nu})$, where $x\in K$ and
$\nu\in\mathbb{Z}$, is the union of precisely $|k_{K}|$ balls of radius
$q_{K}^{-\nu-1}$. For further details, see [3, Chapter 4].

###  2.2. Field extensions

Suppose $(K,|\,\cdot\,|_{K})$ is a non-archimedean local field and $L:K$ is a
finite extension of $K$ of degree $d\in\mathbb{N}$. Then there exists a unique
extension $|\,\cdot\,|_{L}$ of $|\,\cdot\,|_{K}$ to $L$. Furthermore,
$(L,|\,\cdot\,|_{L})$ is also a non-archimedean local field. We say the
extension $L:K$ is totally ramified if the residue class fields $k_{K}$ and
$k_{L}$ are isomorphic. In this case, if $\pi_{K}$ and $\pi_{L}$ are
uniformisers of $K$ and $L$, respectively, then
$|\pi_{K}|_{K}=|\pi_{L}|_{L}^{d}$. Thus,
$\Gamma_{L}=\\{q_{K}^{-\nu/d}:\nu\in\mathbb{Z}\\}$ where
$q_{K}:=|\pi_{K}|_{K}^{-1}$. For further details, see [3, Chapter 7].

To construct a totally ramified extension of $(K,|\,\cdot\,|_{K})$ of an
arbitrary degree $d\in\mathbb{N}$, consider the polynomial $f\in K[X]$ given
by $f(X):=X^{d}-\pi_{K}$. By Eisenstein’s criterion (see [3, Theorem 2.1]),
$f$ is irreducible over $K$. Thus, if $\zeta$ a root of $f$, lying in the
algebraic closure of $K$, then the simple extension $K(\zeta)$ has degree $d$
and is totally ramified by [3, Theorem 7.1].

###  2.3. Vector spaces

Given a valued field $(K,|\,\cdot\,|_{K})$ and $n\in\mathbb{N}$, the
$n$-dimensional vector space $K^{n}$ over $K$ is endowed with the norm

$|\mathbf{x}|_{K}:=\max\\{|x_{1}|_{K},\dots,|x_{n}|_{K}\\}\qquad\textrm{for
all $\mathbf{x}=(x_{1},\dots,x_{n})\in K^{n}$.}$

The ball centred at $\mathbf{x}\in K^{n}$ of radius $r>0$ is then defined by

$B(\mathbf{x},r):=\\{\mathbf{y}\in K^{n}:|\mathbf{y}-\mathbf{x}|_{K}\leq
r\\}.$

###  2.4. Fourier analysis on non-archimedean local fields

By the above discussion, any non-archimedean local field $(K,|\,\cdot\,|_{K})$
is a LCA group and therefore admits an additive Haar measure $\mu$. By
appropriately normalising, one may assume $\mu(\mathfrak{o}_{K})=1$.

Let $\widehat{K}$ denote the Pontryagin dual of $K$. There exists a character
$e\in\widehat{K}$ with the property that the restriction of $e$ to
$\mathfrak{o}_{K}$ is a principal character on the additive subgroup
$\mathfrak{o}_{K}$ whilst the restriction of $e$ to
$\pi_{K}^{-1}\mathfrak{o}_{K}$ is non-principal on the additive subgroup
$\pi_{K}^{-1}\mathfrak{o}_{K}$. We will apply Fourier analysis over the vector
spaces $K^{n}$, which are endowed with the Haar measure given by the $n$-fold
product of $\mu$, also denoted by $\mu$. Given any $\bm{\xi}\in K^{n}$, if one
defines $e_{\bm{\xi}}\colon K^{n}\to\mathbb{C}$ by
$e_{\bm{\xi}}(\mathbf{x}):=e(\mathbf{x}\cdot\bm{\xi})$ for $\mathbf{x}\in
K^{n}$ where $\mathbf{x}\cdot\bm{\xi}:=x_{1}\xi_{1}+\cdots+x_{n}\xi_{n}$, then
$\bm{\xi}\mapsto e_{\bm{\xi}}$ is an isomorphism between $K^{n}$ and
$\widehat{K}^{n}$. For further details see [15, Chapter 1, §8].

Let $\nu$ be a Borel measure on $K^{n}$. By duality, we may also consider this
as a measure on $\widehat{K}^{n}$ (in particular, this applies to the Haar
measure). If $\nu$ is a finite measure, we may define the Fourier transform
and inverse Fourier transform of $\nu$ by

$\widehat{\nu}(\bm{\xi}):=\int_{K^{n}}\overline{e(\mathbf{x}\cdot\bm{\xi})}\,\mathrm{d}\nu(\mathbf{x})\quad\textrm{and}\quad\widecheck{\nu}(\mathbf{x}):=\int_{\widehat{K}^{n}}e(\mathbf{x}\cdot\bm{\xi})\,\mathrm{d}\nu(\bm{\xi})$

With this definition, the rudiments of Fourier analysis such as the inversion
formula, Parseval’s theorem and Plancherel’s theorem hold over $K^{n}$. For
further details see [15, Chapters 2-3].

##  3\. The proof of Proposition 1.2

###  3.1. A structural lemma for sublevel sets

Central to the proof of Proposition 1.2 is a non-archimedean structural
decomposition for sublevel sets of univariate real polynomials due to
Phong–Stein–Sturm [10]. Here we work in the abstract setting of a non-
archimedean local field $(K,|\,\cdot\,|_{K})$. The Phong–Stein–Sturm
decomposition from [10] was extended to such fields in [16], and we state this
version below in Lemma 3.1.

To introduce the key lemma, suppose
$\bm{\xi}=(\xi_{1},...,\xi_{n})\in(\mathfrak{o}_{K})^{n}$ is an $n$-tuple of
roots in the ring of integers $\mathfrak{o}_{K}$ of $K$ and define the monic
polynomial $P_{\bm{\xi}}\in K[X]$ by

$P_{\bm{\xi}}(X)=\prod_{j=1}^{n}(X-\xi_{j}).$ (6)

Given $0<\varepsilon\leq 1$, we are interested in analysing the structure of
the sublevel sets

$\\{z\in\mathfrak{o}_{K}:|P_{\bm{\xi}}(z)|_{K}\leq\varepsilon\\}.$

Naturally, this depends on the distribution of the roots $\xi_{j}$ and, to
understand this, we consider root clusters $\mathcal{C}$, which are simply
defined to be subsets of $\\{\xi_{1},\dots,\xi_{n}\\}$.

######  Lemma 3.1 ([16]).

Suppose $(K,|\,\cdot\,|_{K})$ is a non-archimedean local field and
$\bm{\xi}=(\xi_{1},...,\xi_{n})\in(\mathfrak{o}_{K})^{n}$ is an $n$-tuple of
roots. For all $0<\varepsilon\leq 1$ we have

$\\{z\in\mathfrak{o}_{K}:|P_{\bm{\xi}}(z)|_{K}\leq\varepsilon\\}=\bigcup_{j=1}^{n}B_{K}(\xi_{j};r_{j}(\bm{\xi},\varepsilon))$

where

$r_{j}(\bm{\xi};\varepsilon):=\min_{\mathcal{C}\ni\xi_{j}}\Big{(}\frac{\varepsilon}{\prod_{\xi_{i}\notin\mathcal{C}}|\xi_{j}-\xi_{i}|_{K}}\Big{)}^{1/|\mathcal{C}|}.$
(7)

Here the minimum is taken over all root clusters $\mathcal{C}$ containing
$\xi_{j}$.

######  Remark 3.2.

By taking $\mathcal{C}=\\{\xi_{1},\dots,\xi_{n}\\}$ in the expression defining
the radii in (7), we see that
$r_{j}(\bm{\xi};\varepsilon)\leq\varepsilon^{1/n}$ for $1\leq j\leq n$.

We will work with the following ‘self-referential’ formula for the radii (7).

######  Lemma 3.3.

Let $\bm{\xi}$ and $r_{j}(\bm{\xi};\varepsilon)$ be as in the statement of
Lemma 3.1. For $1\leq j\leq n$ define the root cluster

$\mathcal{C}_{j}:=B_{K}(\xi_{j};r_{j}(\bm{\xi},\varepsilon))\cap\\{\xi_{1},\dots,\xi_{n}\\}.$

Then

$r_{j}(\bm{\xi};\varepsilon)=\Big{(}\frac{\varepsilon}{\prod_{\xi_{i}\notin\mathcal{C}_{j}}|\xi_{j}-\xi_{i}|_{K}}\Big{)}^{1/|\mathcal{C}_{j}|}.$

###### Proof.

Fix $1\leq j\leq n$ and let $\mathcal{C}$ be a root cluster which achieves the
minimum in (7), so that

$r_{j}:=r_{j}(\bm{\xi};\varepsilon)=\Big{(}\frac{\varepsilon}{\prod_{\xi_{i}\notin\mathcal{C}}|\xi_{j}-\xi_{i}|_{K}}\Big{)}^{1/|\mathcal{C}|}.$
(8)

Writing

$\prod_{\xi_{i}\notin\mathcal{C}}|\xi_{j}-\xi_{i}|_{K}=\prod_{\xi_{i}\in\mathcal{C}_{j}\setminus\mathcal{C}}|\xi_{j}-\xi_{i}|_{K}\prod_{\xi_{i}\notin\mathcal{C}_{j}}|\xi_{j}-\xi_{i}|_{K}\prod_{\xi_{i}\in\mathcal{C}\setminus\mathcal{C}_{j}}|\xi_{j}-\xi_{i}|_{K}^{-1}$

and using the fact that $|\xi_{j}-\xi_{i}|_{K}\leq r_{j}$ if and only if
$\xi_{i}\in\mathcal{C}_{j}$, we deduce that

$\prod_{\xi_{i}\notin\mathcal{C}}|\xi_{j}-\xi_{i}|_{K}\leq
r_{j}^{|\mathcal{C}_{j}\setminus\mathcal{C}|-|\mathcal{C}\setminus\mathcal{C}_{j}|}\prod_{\xi_{i}\notin\mathcal{C}_{j}}|\xi_{j}-\xi_{i}|_{K}.$
(9)

Combining (8) and (9) together with the elementary count

$|\mathcal{C}|+|\mathcal{C}_{j}\setminus\mathcal{C}|-|\mathcal{C}\setminus\mathcal{C}_{j}|=|\mathcal{C}_{j}|,$

we obtain

$r_{j}=r_{j}^{(|\mathcal{C}_{j}\setminus\mathcal{C}|-|\mathcal{C}\setminus\mathcal{C}_{j}|)/|\mathcal{C}_{j}|}\Big{(}\frac{\varepsilon}{\prod_{\xi_{i}\notin\mathcal{C}}|\xi_{j}-\xi_{i}|_{K}}\Big{)}^{1/|\mathcal{C}_{j}|}\geq\Big{(}\frac{\varepsilon}{\prod_{\xi_{i}\notin\mathcal{C}_{j}}|\xi_{j}-\xi_{i}|_{K}}\Big{)}^{1/|\mathcal{C}_{j}|}.$

The desired identity immediately follows. ∎

We emphasise that Lemmas 3.1 and 3.3 are valid in any non-archimedean local
field. We will apply them to certain field extensions of the field $K$
appearing in Proposition 1.2.

###  3.2. The main argument

Here we apply the tools introduced in the previous subsection to prove
Proposition 1.2.

###### Proof (of Proposition 1.2).

The argument is broken into steps.

Step 1. Suppose $\mathbf{x}=(x_{1},\dots,x_{n})$,
$\mathbf{y}=(y_{1},\dots,y_{n})\in(\mathfrak{o}_{K})^{n}$ satisfy (3), so that

$|p_{j}(\mathbf{x})-p_{j}(\mathbf{y})|_{K}\leq N^{-n}\qquad\textrm{for $1\leq
j\leq n$,}$

where the $p_{j}\in K[X_{1},\dots,X_{n}]$ are the degree $j$ power sums
$p_{j}(X)=\sum_{\ell=1}^{n}X_{\ell}^{j}$ for $1\leq j\leq n$ and
$N:=(q_{K})^{a}$ for some $a\in\mathbb{N}$. Then $\mathbf{x}$, $\mathbf{y}$
also satisfy

$|e_{j}(\mathbf{x})-e_{j}(\mathbf{y})|_{K}\leq N^{-n}\qquad\textrm{for $1\leq
j\leq n$,}$ (10)

where the $e_{j}\in K[X_{1},\dots,X_{n}]$ are the the degree $j$ elementary
symmetric polynomials $e_{j}(X)=\sum_{k_{1}<\cdots<k_{j}}X_{k_{1}}\cdots
X_{k_{j}}$ for $1\leq j\leq n$. Indeed, this is a direct consequence of the
Girard–Newton formulæ

$je_{j}(X_{1},\ldots,X_{n})=\sum_{i=1}^{j}(-1)^{i-1}e_{j-i}(X_{1},\ldots,X_{n})p_{i}(X_{1},\ldots,X_{n})\qquad\textrm{$1\leq
j\leq n$},$

since the hypothesis $\mathrm{char}\,k_{K}>n$ ensures $|j|_{K}=1$ for $1\leq
j\leq n$.

Step 2. Given an $n$-tuple of roots
$\bm{\xi}=(\xi_{1},\dots,\xi_{n})\in(\mathfrak{o}_{K})^{n}$, define the
polynomial $P_{\bm{\xi}}\in K[X]$ as in (6). In particular,

$P_{\bm{\xi}}(X)=\prod_{j=1}^{n}(X-\xi_{j})=\sum_{j=0}^{n}(-1)^{n-j}e_{n-j}(\bm{\xi})X^{j}$
(11)

Let $K_{\circ}:K$ be a finite extension, and $|\,\cdot\,|_{K_{\circ}}$ the
unique extension of $|\,\cdot\,|_{K}$ to $K_{\circ}$. We can then interpret
$P_{\bm{\xi}}$ as lying in the polynomial ring $K_{\circ}[X]$. Moreover, for
$\mathbf{x}$, $\mathbf{y}$ as in Step 1, it then follows that

$\big{\\{}z\in\mathfrak{o}_{K_{\circ}}:|P_{\mathbf{x}}(z)|_{K_{\circ}}\leq
N^{-n}\big{\\}}=\big{\\{}z\in\mathfrak{o}_{K_{\circ}}:|P_{\mathbf{y}}(z)|_{K_{\circ}}\leq
N^{-n}\big{\\}}.$ (12)

To see this, we note (11), (10) and the ultrametric triangle inequality imply

$|P_{\mathbf{x}}(z)-P_{\mathbf{y}}(z)|_{K_{\circ}}\leq\max_{0\leq j\leq
n}|e_{n-j}(\mathbf{x})-e_{n-j}(\mathbf{y})|_{K}|z|_{K_{\circ}}^{j}\leq
N^{-n}\qquad\textrm{for all $z\in\mathfrak{o}_{K_{\circ}}$.}$

The desired identity (12) now follows from another application of the
ultrametric triangle inequality.

Step 3. In remaining steps we will analyse the structure of the sublevel sets
featured in (12) in order to determine information about $\mathbf{x}$,
$\mathbf{y}$. We will carry out this analysis at two separate scales: a course
scale, introduced here in Step 3, and a finer scale which is analysed in the
remaining steps.

By the Phong–Stein–Sturm sublevel set decomposition from Lemma 3.1, and in
particular the observation in Remark 3.2, for any
$\bm{\xi}=(\xi_{1},\dots,\xi_{n})\in(\mathfrak{o}_{K_{\circ}})^{n}$ we have

$\big{\\{}z\in\mathfrak{o}_{K_{\circ}}:|P_{\bm{\xi}}(z)|_{K_{\circ}}\leq
N^{-n}\big{\\}}\subseteq\bigcup_{j=1}^{n}B_{K_{\circ}}(\xi_{j},N^{-1}).$ (13)

From this and (12), we see that:

  * •

For all $1\leq j\leq n$ there exists some $1\leq j^{\prime}\leq n$ such that
$|x_{j}-y_{j^{\prime}}|_{K}\leq N^{-1}$;

  * •

For all $1\leq j\leq n$ there exists some $1\leq j^{\prime}\leq n$ such that
$|y_{j}-x_{j^{\prime}}|_{K}\leq N^{-1}$.

This sets up a bipartite graph $G=(X,Y,E)$ where the vertex sets
$X:=\\{x_{1},\dots,x_{n}\\}$ and $Y:=\\{y_{1},\dots,y_{n}\\}$ are formed by
the components of $\mathbf{x}$ and $\mathbf{y}$ and $x_{i}\in X$ and $y_{j}\in
Y$ are adjacent if and only if $|x_{i}-y_{j}|_{K}\leq N^{-1}$. It follows from
the above that there are no isolated vertices. Furthermore, the ultrametric
property implies the connected components $G_{1},\dots,G_{M}$ of $G$ are
complete bipartite graphs.

Write $G_{m}=(U_{m},V_{m},E_{m})$ for $1\leq m\leq M$. The vertex sets
$U_{m}\subseteq X$ and $V_{m}\subseteq Y$ are referred to as superclusters. We
let $\alpha_{m}:=|U_{m}|$ and $\beta_{m}:=|V_{m}|$. In light of the above, the
problem is reduced to showing

$\alpha_{m}=\beta_{m}\qquad\textrm{for $1\leq m\leq M$.}$ (14)

Indeed, if this is the case, then we can define a permutation $\sigma$ on
$\\{1,\dots,n\\}$ such that if $x_{j}\in U_{m}$ for some $1\leq j\leq n$ and
$1\leq m\leq M$, then $y_{\sigma(j)}\in V_{m}$. By the properties of the
superclusters, it follows that $|x_{j}-y_{\sigma(j)}|_{K}\leq N^{-1}$ for all
$1\leq j\leq n$.

Step 4. To prove (14), we argue by contradiction. Suppose there exists some
$1\leq m\leq M$ such that $\alpha_{m}\neq\beta_{m}$. By relabelling, we may
assume without loss of generality that $\beta_{1}>\alpha_{1}$ and, moreover,
that $\beta_{1}/\alpha_{1}>1$ maximises the ratio $\beta_{m}/\alpha_{m}$ over
all choices of $1\leq m\leq M$.

We now analyse the problem at a smaller scale, within the superclusters
$U_{m}$ and $V_{m}$. Refining (13), we know from (12) and Lemma 3.1 that

$\bigcup_{j=1}^{n}B_{K_{\circ}}(x_{j},r_{j}(\mathbf{x},N^{-n}))=\bigcup_{j=1}^{n}B_{K_{\circ}}(y_{j},r_{j}(\mathbf{y},N^{-n})).$

Our first observation is that if $x_{j}\in U_{m}$ and $y_{j^{\prime}}\in
V_{m^{\prime}}$ for $m\neq m^{\prime}$, then the balls
$B_{K_{\circ}}(x_{j};r_{j}(\mathbf{x},N^{-n}))$ and
$B_{K_{\circ}}(y_{j^{\prime}};r_{j^{\prime}}(\mathbf{y},N^{-n}))$ are
disjoint. This allows us to home in and analyse the superclusters $U_{1}$,
$V_{1}$ individually.

To simplify notation, for $u\in U_{1}$ and $v\in V_{1}$, write
$r_{X}(u):=r_{i}(\mathbf{x};N^{-n})$ and $r_{Y}(v):=r_{j}(\mathbf{y};N^{-n})$
where $1\leq i,j\leq n$ is such that $u=x_{i}$ and $v=y_{j}$. By the above
observations,

$\bigcup_{u\in U_{1}}B_{K_{\circ}}(u;r_{X}(u))=\bigcup_{v\in
V_{1}}B_{K_{\circ}}(v;r_{Y}(v)).$

We now apply an ultrametric version of the Vitali cover procedure to pass to
disjoint families of balls. In particular, there exist subcollections
$U\subseteq U_{1}$ and $V\subseteq V_{1}$ such that the collections of balls

$\big{\\{}B_{K_{\circ}}(u,r_{X}(u)):u\in
U\big{\\}}\quad\textrm{and}\quad\big{\\{}B_{K_{\circ}}(v,r_{Y}(v)):v\in
V\big{\\}}$

are pairwise disjoint and

$\bigcup_{u\in U}B_{K_{\circ}}(u;r_{X}(u))=\bigcup_{u\in
U_{1}}B_{K_{\circ}}(u;r_{X}(u))=\bigcup_{v\in
V_{1}}B_{K_{\circ}}(v;r_{Y}(v))=\bigcup_{v\in V}B_{K_{\circ}}(v;r_{Y}(v)).$

At this point, we assume our ambient field $K_{\circ}$ is a totally ramified
finite extension of $K$. Under this hypothesis, the residue class field of
$k_{K_{\circ}}$ is isomorphic to $k_{K}$. In particular, since by hypothesis
$|k_{K}|\geq\mathrm{char}\,k_{K}>n$, any ball $B_{K_{\circ}}(x,r)$ cannot be
written as a union of $n$ (not necessarily distinct) balls with strictly
smaller radii. Consequently, $|U|=|V|$ and there exists enumerations of the
sets $U=\\{u_{1},\dots,u_{L}\\}$, $V=\\{v_{1},\dots,v_{L}\\}$ such that

$B_{K_{\circ}}(u_{\ell},r_{X}(u_{\ell}))=B_{K_{\circ}}(v_{\ell},r_{Y}(v_{\ell}))\qquad\textrm{for
$1\leq\ell\leq L$.}$ (15)

At this stage, we wish to conclude that

$r_{X}(u_{\ell})=r_{Y}(v_{\ell})\qquad\textrm{for $1\leq\ell\leq L$.}$ (16)

If we work with $K_{\circ}=K$ in (12), then (16) does not necessarily follow
from (15) owing to the discrete nature of the value group. To address this, we
now further assume that $K_{\circ}\colon K$ is a degree $n!$ totally ramified
extension. Under this hypothesis, the value groups $\Gamma_{K}$ and
$\Gamma_{L}$ take the form

$\Gamma_{K}=\\{q_{K}^{-\nu}:\nu\in\mathbb{Z}\\}\qquad\textrm{and}\qquad\Gamma_{K_{\circ}}=\\{q_{K}^{-\nu/n!}:\nu\in\mathbb{Z}\\}.$

In particular, $\Gamma_{K_{\circ}}$ contains the quantities $r_{X}(u_{\ell})$
and $r_{Y}(v_{\ell})$. Thus, working over $\mathfrak{o}_{K_{\circ}}$, we may
deduce (16) from (15).

Step 5. We now apply the self-referential form of the Phong–Stein–Sturm
sublevel decomposition from Lemma 3.3 to obtain a formula for the radii
appearing in (16). For $1\leq\ell\leq L$, let

$\mathcal{C}_{X}(u_{\ell}):=B_{K_{\circ}}(u_{\ell},r_{X}(u_{\ell}))\cap
X\quad\textrm{and}\quad\mathcal{C}_{Y}(v_{\ell}):=B_{K_{\circ}}(v_{\ell},r_{Y}(v_{\ell}))\cap
Y$

denote the clusters appearing in Lemma 3.3, which realise the minimum in (7).

We first consider the contributions to the radii arising from roots in
superclusters other than $U_{1}$ and $V_{1}$. By the ultrametric property, for
each $2\leq m\leq M$ there exists some $D_{m}>N^{-1}$ such that

$|u_{\ell}-u^{\prime}|_{K}=|v_{\ell}-v^{\prime}|_{K}=D_{m}\qquad\textrm{for
$1\leq\ell\leq L$ and $u^{\prime}\in U_{m}$, $v^{\prime}\in V_{m}$.}$

Consequently, recalling the definition of the $\alpha_{m}$ and $\beta_{m}$
from Step 3, we have

$\prod_{u^{\prime}\in X\setminus
U_{1}}|u_{\ell}-u^{\prime}|_{K}=\prod_{m=2}^{M}D_{m}^{\alpha_{m}}\quad\textrm{and}\quad\prod_{v^{\prime}\in
Y\setminus V_{1}}|v_{\ell}-v^{\prime}|_{K}=\prod_{m=2}^{M}D_{m}^{\beta_{m}}.$
(17)

We now turn to the contributions of roots within $U_{1}$ and $V_{1}$. For
$1\leq\ell,\ell^{\prime}\leq L$ with $\ell\neq\ell^{\prime}$ we have

$|u_{\ell}-u^{\prime}|_{K}=|u_{\ell}-u_{\ell^{\prime}}|_{K}=|v_{\ell}-v_{\ell^{\prime}}|_{K}=|v_{\ell}-v^{\prime}|_{K}\quad\textrm{for
all $u^{\prime}\in\mathcal{C}_{X}(u_{\ell^{\prime}})$,
$v^{\prime}\in\mathcal{C}_{Y}(v_{\ell^{\prime}})$.}$

In particular, if we define $s_{\ell}:=|\mathcal{C}_{X}(u_{\ell})|$ and
$t_{\ell}:=|\mathcal{C}_{Y}(v_{\ell})|$ for $1\leq\ell\leq L$, it follows that

$\prod_{\begin{subarray}{c}u^{\prime}\in U_{1}\\\
u^{\prime}\notin\mathcal{C}_{X}(u_{\ell})\end{subarray}}|u_{\ell}-u^{\prime}|_{K}=\prod_{\begin{subarray}{c}1\leq\ell^{\prime}\leq
L\\\
\ell^{\prime}\neq\ell\end{subarray}}|u_{\ell}-u_{\ell^{\prime}}|_{K}^{s_{\ell^{\prime}}}\quad\textrm{and
}\prod_{\begin{subarray}{c}v^{\prime}\in V_{1}\\\
v^{\prime}\notin\mathcal{C}_{Y}(v_{\ell})\end{subarray}}|v_{\ell}-v^{\prime}|_{K}=\prod_{\begin{subarray}{c}1\leq\ell^{\prime}\leq
L\\\
\ell^{\prime}\neq\ell\end{subarray}}|v_{\ell}-v_{\ell^{\prime}}|_{K}^{t_{\ell^{\prime}}}$
(18)

whilst we also have

$s_{1}+\cdots+s_{L}=\alpha_{1}<\beta_{1}=t_{1}+\cdots+t_{L}.$ (19)

Combining Lemma 3.3 with (17) and (18), and applying the identity (15), we
conclude that

$\bigg{(}\frac{N^{-n}}{\prod_{\begin{subarray}{c}1\leq\ell^{\prime}\leq L\\\
\ell^{\prime}\neq\ell\end{subarray}}|u_{\ell}-u_{\ell^{\prime}}|_{K}^{s_{\ell^{\prime}}}\prod_{m=2}^{M}D_{m}^{\alpha_{m}}}\bigg{)}^{1/s_{\ell}}=\bigg{(}\frac{N^{-n}}{\prod_{\begin{subarray}{c}1\leq\ell^{\prime}\leq
L\\\
\ell^{\prime}\neq\ell\end{subarray}}|v_{\ell}-v_{\ell^{\prime}}|_{K}^{t_{\ell^{\prime}}}\prod_{m=2}^{M}D_{m}^{\beta_{m}}}\bigg{)}^{1/t_{\ell}}$
(20)

for all $1\leq\ell\leq L$. Thus, raising the above display to the
$s_{\ell}t_{\ell}$ power and rearranging the resulting expression gives

$N^{-n(t_{\ell}-s_{\ell})}\prod_{\begin{subarray}{c}1\leq\ell^{\prime}\leq
L\\\
\ell^{\prime}\neq\ell\end{subarray}}|u_{\ell}-u_{\ell^{\prime}}|_{K}^{-(s_{\ell^{\prime}}t_{\ell}-s_{\ell}t_{\ell^{\prime}})}=\prod_{m=2}^{M}D_{m}^{t_{\ell}\alpha_{m}-s_{\ell^{\prime}}\beta_{m}}.$
(21)

Taking the product of either side of the identity (21) over all choices of
$\ell$, we deduce from (19) that

$N^{-n(\beta_{1}-\alpha_{1})}\prod_{\begin{subarray}{c}1\leq\ell,\ell^{\prime}\leq
L\\\
\ell^{\prime}\neq\ell\end{subarray}}|u_{\ell}-u_{\ell^{\prime}}|_{K}^{-(s_{\ell^{\prime}}t_{\ell}-s_{\ell}t_{\ell^{\prime}})}=\prod_{m=2}^{M}D_{m}^{\beta_{1}\alpha_{m}-\alpha_{1}\beta_{m}}$

and therefore, by parity considerations,

$N^{-n(\beta_{1}-\alpha_{1})}=\prod_{m=1}^{M}D_{j}^{\beta_{1}\alpha_{m}-\alpha_{1}\beta_{m}}.$

From our labelling of the superclusters, we know
$\beta_{1}/\alpha_{1}\geq\beta_{m}/\alpha_{m}$ for all $1\leq m\leq M$.
Furthermore, since $\alpha_{1}+\cdots+\alpha_{M}=\beta_{1}+\cdots+\beta_{M}=n$
and $\beta_{1}/\alpha_{1}>1$, there must exist at least one choice of $m$ for
which $\beta_{1}/\alpha_{1}>\beta_{m}/\alpha_{m}$ (that is, the inequality is
strict). Consequently, all of the exponents
$\beta_{1}\alpha_{m}-\alpha_{1}\beta_{m}$ are non-negative and at least one
exponent is strictly positive. Thus, since $D_{m}>N^{-1}$ for $1\leq m\leq M$,
we conclude that

$N^{-n(\beta_{1}-\alpha_{1})}>N^{-n(\beta_{1}-\alpha_{1})},$

which is a contradiction. This arises from the assumption that (14) fails, and
so (14) must hold, concluding the proof. ∎

##  4\. The Córdoba–Fefferman argument

In this section we apply the standard Córdoba–Fefferman argument [5] in order
to obtain Theorem 1.1 from Proposition 1.2.

###### Proof (of Theorem 1.1).

By translation invariance, we may assume $\mathbf{x}=0$, Letting
$\delta:=q_{K}^{-\alpha}$ and $\varphi:=\chi_{B_{\delta^{-n}}}$ denote the
characteristic function of the ball $B_{\delta^{-n}}:=B(0,q_{K}^{\alpha n})$,
we have

$|Ef|^{2m}\cdot\varphi=\sum_{\begin{subarray}{c}I_{j},J_{j}\in\mathcal{I}(\delta)\\\
1\leq j\leq
m\end{subarray}}\prod_{j=1}^{m}Ef_{I_{j}}\cdot\varphi\;\overline{\prod_{j=1}^{m}Ef_{J_{j}}\cdot\varphi}.$

Thus, by Parseval’s theorem,

$\|Ef\|_{L^{2m}(B_{\delta^{-n}})}^{2m}=\sum_{\begin{subarray}{c}I_{j},J_{j}\in\mathcal{I}(\delta)\\\
1\leq j\leq
m\end{subarray}}\int_{K^{n}}\Big{(}\prod_{j=1}^{m}Ef_{I_{j}}\cdot\varphi\Big{)}\;\widehat{}\;(\bm{\xi})\;\overline{\Big{(}\prod_{j=1}^{m}Ef_{J_{j}}\cdot\varphi\Big{)}\;\widehat{}\;(\bm{\xi})}\,\mathrm{d}\mu(\bm{\xi}).$
(22)

Let $\nu$ denote the pushforward of the Haar measure on $\mathfrak{o}_{K}$
under the moment mapping $\gamma\colon\mathfrak{o}_{K}\to\mathfrak{o}_{K}^{n}$
given by $\gamma(t):=(t,t^{2},\dots,t^{n})$ for all $t\in\mathfrak{o}_{K}$.
Observe that

$Eg=(\,g\mathrm{d}\nu\,)\;\widecheck{}\;\qquad\textrm{for $g\in
L^{1}(\mathfrak{o}_{K})$}$

and so $(Ef_{I}\cdot\varphi)\;\widehat{}\;=\widehat{\varphi}\;\ast
f_{I}\mathrm{d}\nu$ for any $I\in\mathcal{I}(\delta)$. Thus, fixing $I_{j}$,
$J_{j}\in\mathcal{I}(\delta)$ for $1\leq j\leq n$, it follows that the right-
hand integrand in (22) can be written as

$\big{(}\widehat{\varphi}\;\ast
f_{I_{1}}\mathrm{d}\nu\big{)}\ast\cdots\ast\big{(}\widehat{\varphi}\;\ast
f_{I_{m}}\mathrm{d}\nu\big{)}(\bm{\xi})\;\overline{\big{(}\widehat{\varphi}\;\ast
f_{J_{1}}\mathrm{d}\nu\big{)}\ast\cdots\ast\big{(}\widehat{\varphi}\;\ast
f_{J_{n}}\mathrm{d}\nu\big{)}(\bm{\xi})}.$ (23)

By a simple computation,
$\widehat{\varphi}=\delta^{-n^{2}}\chi_{B(0,\delta^{n})}$ and, in particular,

$\mathrm{supp}\,(\widehat{\varphi}\ast
f_{I}\mathrm{d}\nu)\subseteq\big{\\{}\bm{\xi}\in\widehat{K}^{n}:|\bm{\xi}-\gamma(s)|_{K}\leq\delta^{n}\textrm{
for some $s\in I$}\big{\\}}\qquad\textrm{for $I\in\mathcal{I}(\delta)$}.$

Moreover, if $\bm{\xi}\in\widehat{K}^{n}$ lies in the support of the function
in (23), then

$\Big{|}\bm{\xi}-\sum_{j=1}^{m}\gamma(s_{j})\Big{|}_{K}\leq\delta^{n}\textrm{
and
}\Big{|}\bm{\xi}-\sum_{j=1}^{m}\gamma(t_{j})\Big{|}_{K}\leq\delta^{n}\textrm{
for some $s_{j}\in I_{j}$, $t_{j}\in J_{j}$, $1\leq j\leq m$.}$

Now suppose the support of the function in (23) is non-empty for some choice
of $I_{j}$, $J_{j}\in\mathcal{I}(\delta)$ for $1\leq j\leq m$. By the
preceding observations, there must exist $s_{j}\in I_{j}$, $t_{j}\in J_{j}$
for $1\leq j\leq m$ such that

$\Big{|}\sum_{j=1}^{m}\gamma(s_{j})-\sum_{j=1}^{m}\gamma(t_{j})\Big{|}_{K}\leq
q_{K}^{-\alpha n}.$

Applying Proposition 1.2, there exists a permutation $\sigma$ on
$\\{1,\cdots,m\\}$ such that $|t_{j}-s_{\sigma(j)}|_{K}\leq q_{K}^{-\alpha}$
for all $1\leq j\leq m$. By the ultrametric property, this can only happen if
$J_{j}=I_{\sigma(j)}$ for all $1\leq j\leq m$.

In light of the discussion of the previous paragraph, we see that all the
‘off-diagonal’ terms of the right-hand sum in (22) are zero and, in
particular,

$\displaystyle\|Ef\|_{L^{2m}(B_{\delta^{-n}})}^{2m}$ $\displaystyle\leq
m!\sum_{I_{1},\dots,I_{m}\in\mathcal{I}(\delta)}\int_{K^{n}}\Big{|}\Big{(}\prod_{j=1}^{m}Ef_{I_{j}}\cdot\varphi\Big{)}\;\widehat{}\;(\bm{\xi})\;\Big{|}^{2}\,\mathrm{d}\mu(\bm{\xi})$
$\displaystyle\leq
m!\Big{\|}\big{(}\sum_{I\in\mathcal{I}(\delta)}|Ef_{I}|^{2}\big{)}^{1/2}\Big{\|}_{L^{2k}(B_{\delta^{-n}})}^{2m}$

where the second identity is a consequence of Plancherel’s theorem. This
concludes the proof. ∎

##  Appendix A Adapting the argument to archimedean local fields

###  A.1. Key ingredients

In this section we sketch how the arguments of §3 can be adapted to work in
the archimedean setting. The main result is as follows.

######  Proposition A.1.

Let $(K,|\,\cdot\,|_{K})$ be an archimedean local field and $n\in\mathbb{N}$.
There exists a constant $C_{n}\geq 1$, depending only on $n$, such that the
following holds. Let $N\geq 1$ and suppose $(x_{1},\dots,x_{n})$,
$(y_{1},\dots,y_{n})\in B_{K}(0,1)^{n}$ satisfy

$|x_{1}^{j}+\cdots+x_{n}^{j}-y_{1}^{j}-\cdots-y_{n}^{j}|_{K}\leq
N^{-n}\qquad\textrm{for $1\leq j\leq n$.}$

Then there exists a permutation $\sigma$ on $\\{1,\cdots,n\\}$ such that
$|x_{j}-y_{\sigma(j)}|_{K}\leq C_{n}N^{-1}$ for all $1\leq j\leq n$.

Proposition A.1 can be combined with the Córdoba–Fefferman argument described
in §4 to yield the analogue of Theorem 1.1 for archimedean local fields. The
proof of Proposition A.1 closely follows that of Proposition 1.2, albeit with
a few minor points of divergence. Here we review the key tools used in the
argument and how they differ from those used in the non-archimedean setting.

The Phong–Stein–Sturm decomposition. An approximate version of Lemma 3.1 holds
over archimedean local fields. Moreover, if $(K,|\,\cdot\,|_{K})$ is any
valued field with non-trivial absolute value and
$\bm{\xi}=(\xi_{1},...,\xi_{n})\in B_{K}(0,1)^{n}$ is an $n$-tuple of roots,
then for all $0<\varepsilon\leq 1$ we have

$\bigcup_{j=1}^{n}B_{K}(\xi_{j};2^{-n}r_{j}(\bm{\xi},\varepsilon))\subseteq\\{z\in
K:|P_{\bm{\xi}}(z)|\leq\varepsilon\\}\subseteq\bigcup_{j=1}^{n}B_{K}(\xi_{j};2^{n}r_{j}(\bm{\xi},\varepsilon));$

see [8, Proposition 3.3]. We also note that for any $\lambda\geq 1$ we have

$\lambda^{1/n}r_{j}(\bm{\xi},\varepsilon)\leq
r_{j}(\bm{\xi},\lambda\varepsilon)\leq\lambda r_{j}(\bm{\xi},\varepsilon).$

Using the above observations, Steps 1 - 3 in the proof of Proposition 1.2 can
be carried over in a straightforward manner to the archimedean setting, with
additional constant factors appearing throughout the argument. In contrast
with the non-archimedean case, we work directly over the field $K$ rather than
some field extension (indeed, no rich theory of field extensions is available
in the archimedean setting). The superclusters are defined using the condition
$|x-y|_{K}\leq\rho N^{-1}$, where $\rho\geq 1$ is a parameter which is chosen
large, depending only on $n$, so as to force a contradiction at the end of the
argument.

A Vitali-type covering lemma. Step 4 of the proof of Proposition 1.2 featured
an application of the ultrametric Vitali covering lemma, which was used to
pass to the two identical families of balls in (15). The ultrametric covering
lemma is very clean, owing to the fact that any two balls in an ultrametric
space are either nested or disjoint. To adapt the argument to the archimedean
setting, we make use of the following somewhat technical variant of the
original Vitali covering lemma.

######  Lemma A.2.

Let $\mathcal{B}_{X}$, $\mathcal{B}_{Y}$ be finite sets of closed balls in
$\mathbb{R}^{d}$ of cardinality at most $n\in\mathbb{N}$. Suppose that
$\lambda\geq 1$ is such that

$\bigcup_{B_{X}\in\mathcal{B}_{X}}B_{X}\subseteq\bigcup_{B_{Y}\in\mathcal{B}_{Y}}\lambda\cdot
B_{Y}\quad\textrm{and}\quad\bigcup_{B_{Y}\in\mathcal{B}_{Y}}B_{Y}\subseteq\bigcup_{B_{X}\in\mathcal{B}_{X}}\lambda\cdot
B_{X}.$ (24)

Then there exist
$\mathcal{B}_{X}^{\prime}=\\{B_{X}^{1},\dots,B_{X}^{L}\\}\subseteq\mathcal{B}_{X}$
and
$\mathcal{B}_{Y}^{\prime}=\\{B_{Y}^{1},\dots,B_{Y}^{L}\\}\subseteq\mathcal{B}_{Y}$
and a constant $R=R(n,\lambda)$ such that the following hold:

  1. 1)

Strong separation. For all $1\leq\ell<\ell^{\prime}\leq L$, we have

$2R\cdot B_{X}^{\ell}\cap 2R\cdot
B_{X}^{\ell^{\prime}}=\emptyset\quad\textrm{and}\quad 2R\cdot B_{Y}^{\ell}\cap
2R\cdot B_{Y}^{\ell^{\prime}}=\emptyset.$

  2. 2)

Vitali covering.

$\displaystyle\bigcup_{B_{X}\in\mathcal{B}_{X}}B_{X}\subseteq\bigcup_{\ell=1}^{L}R\cdot
B_{X}^{\ell}\quad\textrm{and}\quad\bigcup_{B_{Y}\in\mathcal{B}_{Y}}B_{Y}\subseteq\bigcup_{\ell=1}^{L}R\cdot
B_{Y}^{\ell}$

  3. 3)

Comparable balls. For all $1\leq\ell\leq L$, we have

$B_{X}^{\ell}\subseteq R\cdot B_{Y}^{\ell}\quad\textrm{and}\quad
B_{Y}^{\ell}\subseteq R\cdot B_{X}^{\ell}.$

Here, given a ball $B\subseteq\mathbb{R}^{d}$ and $\lambda>0$, we let
$\mathrm{rad}\,B$ denote the radius of $B$ and $\lambda\cdot B$ denote the
ball concentric to $B$ but with radius $\lambda\mathrm{rad}\,B$. In applying
the lemma, we identify the archimedean local field $\mathbb{C}$ with the
metric space $\mathbb{R}^{2}$.

######  Remark A.3.

A key feature of Lemma A.2 is that the parameter $R$ is allowed to depend on
the number of balls $n$ (in stark contrast with the classical Vitali covering
lemma). This flexibility allows for the comparability between the balls
$B_{X}^{\ell}$ and $B_{Y}^{\ell}$. It also allows for the strong separation
property in 1), where the separation parameter $2R$ is larger than the
dilation parameter in 2).

The comparability property 3) is a surrogate for the identification between
balls in (15) in the archimedean setting. Similarly, the strong separation
property 1) is used to establish approximate versions of the identities in
(18).

Since Lemma A.2 is a new feature of the argument, we present the full proof in
§A.2 below.

The self-referential formula for the radii. The final ingredient we highlight
from the proof of Proposition 1.2 is the self-referential formula for the
radii from Lemma 3.3; recall, this is used to establish the identity (20) in
Step 5. The proof of Lemma 3.3 does not rely on the ultrametric triangle
inequality, and the result remains valid as stated in any valued field
$(K,|\,\cdot\,|_{K})$ with non-trivial absolute value. However, for the proof
of Proposition A.1 we require a slight extension of the formula. For $1\leq
j\leq n$ and $\lambda\geq 1$ define the root cluster

$\mathcal{C}_{j,\lambda}:=B_{K}(\xi_{j};\lambda
r_{j}(\bm{\xi},\varepsilon))\cap\\{\xi_{1},\dots,\xi_{n}\\}.$

Then the proof of Lemma 3.3 shows that

$r_{j}(\bm{\xi};\varepsilon)\leq\Big{(}\frac{\varepsilon}{\prod_{\xi_{i}\notin\mathcal{C}_{j,\lambda}}|\xi_{j}-\xi_{i}|_{K}}\Big{)}^{1/|\mathcal{C}_{j,\lambda}|}\leq\lambda
r_{j}(\bm{\xi};\varepsilon).$

The approximate formula can be used to establish an approximate version of
(20).

###  A.2. Proof of the Vitali-type lemma

In this section, we prove Lemma A.2. The first step is the following simple
consequence of the classical Vitali covering lemma.

######  Lemma A.4.

Let $\mathcal{B}$ be a finite collection of balls in $\mathbb{R}^{d}$ of
cardinality at most $n$ and $R\geq 1$. Then there exists a subcollection
$\mathcal{B}\subseteq\mathcal{B}$ and a constant $\lambda=\lambda(n,R)\geq 1$
depending only on $n$ and $R$ such that

  1. 1)

Strong separation. The large dilates $\\{\lambda R\cdot
B^{\prime}:B^{\prime}\in\mathcal{B}^{\prime}\\}$ are pairwise disjoint.

  2. 2)

Vitali covering. The small dilates $\\{\lambda\cdot
B^{\prime}:B^{\prime}\in\mathcal{B}^{\prime}\\}$ form a Vitali cover in the
sense that

$\bigcup_{B\in\mathcal{B}}B\subseteq\bigcup_{B^{\prime}\in\mathcal{B}^{\prime}}\lambda\cdot
B^{\prime}.$

###### Proof.

The proof is based on repeated application of the classical Vitali covering
lemma and pigeonholing. Starting with $\mathcal{B}_{0}:=\mathcal{B}$, we
recursively construct a chain of proper subsets
$\mathcal{B}_{m}\subset\mathcal{B}_{m-1}\subset\dots\subset\mathcal{B}_{0}$
such that

$\bigcup_{B\in\mathcal{B}}B\subseteq\bigcup_{B_{m}\in\mathcal{B}_{m}}\lambda_{m}\cdot
B$ (25)

where $\lambda_{m}:=(3R)^{m}$.

Suppose that $\mathcal{B}_{m}$ has already been constructed and satisfies
(25). Apply the classical Vitali covering lemma to the collection of dilated
balls $\\{R\lambda_{m}\cdot B:B\in\mathcal{B}_{m}\\}$ to obtain a
subcollection $\mathcal{B}_{m+1}\subseteq\mathcal{B}_{m}$ such that

$\\{R\lambda_{m}B:B\in\mathcal{B}_{m+1}\\}\quad\textrm{are pairwise disjoint}$

and, noting $\lambda_{m+1}=3R\lambda_{m}$,

$\bigcup_{B\in\mathcal{B}}B\subseteq\bigcup_{B\in B_{m}}R\lambda_{m}\cdot
B\subseteq\bigcup_{B\in\mathcal{B}_{m+1}}\lambda_{m+1}\cdot B.$

If $\mathcal{B}_{m+1}=\mathcal{B}_{m}$, then the algorithm terminates;
otherwise, $\mathcal{B}_{m+1}\subset\mathcal{B}_{m}$ is a proper subset, as
required.

By pigeonholing, the algorithm must terminate after at most $n-1$ steps. If
$0\leq M\leq n-1$ is the terminal step, then the desired properties hold with
$\mathcal{B}^{\prime}:=\mathcal{B}_{M}$ and $\lambda:=\lambda_{M}$. ∎

We now turn to the proof of Lemma A.2. For a pair of balls $B_{1}$,
$B_{2}\subseteq\mathbb{R}^{d}$ we frequently make use of the following
consequence of triangle inequality:

If $B_{1}\cap B_{2}\neq\emptyset$, then $B_{1}\subseteq\lambda\cdot B_{2}$
where $\mathrm{rad}\,(\lambda\cdot
B_{2})=2\mathrm{rad}\,B_{1}+\mathrm{rad}\,B_{2}$. (26)

Note, in particular, that the dilate $\lambda\cdot B_{2}$ in the above display
satisfies

$\mathrm{rad}\,(\lambda\cdot B_{2})\leq
3\max\\{\mathrm{rad}\,B_{1},\mathrm{rad}\,B_{2}\\}.$

###### Proof (of Lemma A.2).

We first note that it suffices to construct families
$\mathcal{B}_{X}^{\prime}$ and $\mathcal{B}_{Y}^{\prime}$ satisfying
properties 2) and 3) only. Indeed, once this is achieved, one may apply Lemma
A.4 to pass subcollections of $\mathcal{B}_{X}^{\prime}$ and
$\mathcal{B}_{Y}^{\prime}$ which satisfy 1) in addition to 2) and 3), with a
larger (but nevertheless still admissible) choice of $R$. More precisely, we
first apply Lemma A.4 to, say, the collection $\mathcal{B}_{X}^{\prime}$ (or
balls obtained by suitably dilating the
$B_{X}^{\prime}\in\mathcal{B}_{X}^{\prime}$) to pass to a subcollection which
satisfies the strong separation. One can then pass to a suitable subcollection
of $\mathcal{B}_{Y}^{\prime}$ by using the comparability property 3).

We now turn to the task of constructing the sets $\mathcal{B}_{X}^{\prime}$
and $\mathcal{B}_{Y}^{\prime}$ satisfying 2) and 3). To this end, we will
construct a sequence of balls $B_{X}^{1},\dots,B_{X}^{\ell}\in\mathcal{B}_{X}$
and $B_{Y}^{1},\dots,B_{Y}^{\ell}\in\mathcal{B}_{Y}$ and a sequence of
constants $C_{\ell}\geq\cdots\geq C_{1}\geq 1$ using a recursive algorithm. In
particular, defining

$\displaystyle\mathcal{B}_{X,\ell}$
$\displaystyle:=\\{B_{X}\in\mathcal{B}_{X}:B_{X}\cap C_{k}\cdot
B_{X}^{k}=\emptyset\textrm{ for all $1\leq k\leq\ell$}\\},$ (27)
$\displaystyle\mathcal{B}_{Y,\ell}$
$\displaystyle:=\\{B_{Y}\in\mathcal{B}_{Y}:B_{Y}\cap C_{k}\cdot
B_{Y}^{k}=\emptyset\textrm{ for all $1\leq k\leq\ell$}\\},$

these objects have the following properties:

1) Strong separation. Let $\rho\geq 1$ be a fixed parameter, chosen suitably
large depending only on $n$ and $\lambda$ to satisfy the forthcoming
requirements of the proof.

  1. 1)X,ℓ

If $B_{X}\in\mathcal{B}_{X,\ell}$, then $B_{X}\cap\rho C_{k}\cdot
B_{X}^{k}=\emptyset$ for $1\leq k\leq\ell$,

  2. 1)${}_{\,Y,\ell}$

If $B_{Y}\in\mathcal{B}_{Y,\ell}$, then $B_{Y}\cap\rho C_{k}\cdot
B_{Y}^{k}=\emptyset$ for $1\leq k\leq\ell$.

This condition will play a minor technical role in the proof.

2) Vitali condition. Let $C=C(n,\lambda):=\lambda n^{1/d}$.

  1. 2)X,ℓ

If $B_{X}\in\mathcal{B}_{X,k-1}$, then $\mathrm{rad}\,B_{X}\leq
C\mathrm{rad}\,B_{X}^{k}$ for $1\leq k\leq\ell$;

  2. 2)${}_{\,Y,\ell}$

If $B_{Y}\in\mathcal{B}_{Y,k-1}$, then $\mathrm{rad}\,B_{Y}\leq
C\mathrm{rad}\,B_{Y}^{k}$ for $1\leq k\leq\ell$;

3) Comparable balls. Let $\bar{C}=\bar{C}(n,\lambda):=2C+1$.

  1. 3)ℓ

For all $1\leq k\leq\ell$, we have

$B_{X}^{k}\subseteq\bar{C}\cdot B_{Y}^{k}\quad\textrm{and}\quad
B_{Y}^{k}\leq\bar{C}\cdot B_{X}^{k}.$

Suppose $B_{X}^{1},\dots,B_{X}^{\ell}\subseteq\mathcal{B}_{X}$,
$B_{Y}^{1},\dots,B_{Y}^{\ell}\subseteq\mathcal{B}_{Y}$ and
$(C_{k})_{k=1}^{\ell}$ have already been constructed and satisfy the
properties listed above.

Stopping condition. If either $\mathcal{B}_{X,\ell}=\emptyset$ or
$\mathcal{B}_{Y,\ell}=\emptyset$, then the algorithm terminates.

Recursive step. Suppose the algorithm has not terminated at step $\ell$ so
that $\mathcal{B}_{X,\ell}\neq\emptyset$ and
$\mathcal{B}_{Y,\ell}\neq\emptyset$. Let
$B_{X}^{\ell+1,*}\in\mathcal{B}_{X,\ell}$ and
$B_{Y}^{\ell+1,*}\in\mathcal{B}_{Y,\ell}$ be balls of maximal radii lying in
these sets.

By symmetry, we may assume that
$\mathrm{rad}\,B_{X}^{\ell+1,*}\geq\mathrm{rad}\,B_{Y}^{\ell+1,*}$. In this
case, we define $B_{X}^{\ell+1}:=B_{X}^{\ell+1,*}$, so that Property 2)X,ℓ+1
clearly holds.

We claim that

$B_{X}^{\ell+1}\subseteq\bigcup_{B_{Y}\in\mathcal{B}_{Y,\ell}}\lambda\cdot
B_{Y}.$ (28)

Indeed, suppose the above inclusion fails so that, by the hypothesis (24),
there exists some $B_{Y}\in\mathcal{B}_{Y}\setminus\mathcal{B}_{Y,\ell}$ such
that

$B_{X}^{\ell+1}\cap\lambda\cdot B_{Y}\neq\emptyset.$ (29)

Since $B_{Y}\in\mathcal{B}_{Y}\setminus\mathcal{B}_{Y,\ell}$, there exists
some $1\leq k\leq\ell$ such that

$B_{Y}\cap C_{k}\cdot B_{Y}^{k}\neq\emptyset.$ (30)

We choose $k$ to be minimal with this property. Thus, $B_{Y}\cap C_{j}\cdot
B_{Y}^{j}=\emptyset$ for all $1\leq j\leq k-1$, which is precisely the
condition $B_{Y}\in\mathcal{B}_{Y,k-1}$. Consequently, by Property
2)${}_{\,Y,k}$, we have $\mathrm{rad}\,B_{Y}\leq C\mathrm{rad}\,B_{Y}^{k}$.
Recalling (30) and applying the triangle inequality in the form of (26)
together with Property 3)ℓ, we see that

$\lambda\cdot B_{Y}\subseteq(\rho/2)C_{k}\cdot B_{Y}^{k}\subseteq\rho
C_{k}\cdot B_{X}^{k},$ (31)

provided $\rho$ is suitably chosen. Combining (29) and (31), we have

$B_{X}^{\ell+1}\cap\rho C_{k}\cdot B_{X}^{k}\neq\emptyset.$

By Property 1)${}_{\,X,\ell}$, it follows that
$B_{X}^{\ell+1}\notin\mathcal{B}_{X,\ell}$, but this contradicts our choice of
$B_{X}^{\ell+1}$.

In view of (28), we fix some $B_{Y}^{\ell+1}\in\mathcal{B}_{Y,\ell}$ such that
$\lambda\cdot B_{Y}^{\ell+1}$ has non-trivial intersection with
$B_{X}^{\ell+1}$ with maximal possible radius. It follows that

$\mathrm{rad}\,B_{Y}^{\ell+1}\leq\mathrm{rad}\,B_{Y}^{\ell+1,*}\leq\mathrm{rad}\,B_{X}^{\ell+1,*}=\mathrm{rad}\,B_{X}^{\ell+1},$
(32)

whilst

$\mathcal{L}^{d}\big{(}B_{X}^{\ell+1}\big{)}\leq\sum_{\begin{subarray}{c}B_{Y}\in\mathcal{B}_{Y,\ell}\\\
\lambda\cdot B_{Y}\cap
B_{X}^{\ell+1}\neq\emptyset\end{subarray}}\mathcal{L}^{d}\big{(}\lambda\cdot
B_{Y}\big{)}\leq C^{d}\mathcal{L}^{d}\big{(}B_{Y}^{\ell+1}\big{)},$

where $\mathcal{L}^{d}$ denotes the $d$-dimensional Lebesgue measure. Thus,
$\mathrm{rad}\,B_{X}^{\ell+1}\leq CB_{Y}^{\ell+1}$. Combining these
observations with (26) establishes Property 3)ℓ+1. Similarly, arguing as in
(32), given $B_{Y}\in\mathcal{B}_{Y,\ell}$, it follows that
$\mathrm{rad}\,B_{Y}\leq\mathrm{rad}\,B_{X}^{\ell+1}$. Combining this with
Property 3)ℓ+1, we have $\mathrm{rad}\,B_{Y}\leq
C\mathrm{rad}\,B_{Y}^{\ell+1}$, and so Property 2)${}_{\,Y,\ell+1}$ also
holds.

It remains to construct the constant $C_{\ell+1}$ and verify 1)X,ℓ+1 and
1)${}_{\,Y,\ell+1}$. For $0\leq m\leq 2n+1$ consider the sets

$\displaystyle\mathcal{B}_{X,\ell+1}^{(m)}$
$\displaystyle:=\mathcal{B}_{X,\ell}\cap\\{B_{X}\in\mathcal{B}_{X}:B_{X}\cap
C_{\ell}\rho^{m}\cdot B_{X}^{\ell+1}=\emptyset\\},$
$\displaystyle\mathcal{B}_{Y,\ell+1}^{(m)}$
$\displaystyle:=\mathcal{B}_{Y,\ell}\cap\\{B_{Y}\in\mathcal{B}_{Y}:B_{Y}\cap
C_{\ell}\rho^{m}\cdot B_{Y}^{\ell+1}=\emptyset\\},$

By a pigeonholing argument, there must exist some choice of $0\leq m\leq 2n$
such that

$\mathcal{B}_{X,\ell+1}^{(m+1)}=\mathcal{B}_{X,\ell+1}^{(m)}\quad\textrm{and}\quad\mathcal{B}_{Y,\ell+1}^{(m+1)}=\mathcal{B}_{Y,\ell+1}^{(m)}.$
(33)

With this fixed value of $m$, define $C_{\ell+1}:=C_{\ell}\rho^{m}$ so that
$\mathcal{B}_{X,\ell+1}=\mathcal{B}_{X,\ell+1}^{(m)}$ and
$\mathcal{B}_{Y,\ell+1}=\mathcal{B}_{Y,\ell+1}^{(m)}$. It immediately follows
from (33) and 1)X,ℓ and 1)${}_{\,Y,\ell}$ that 1)X,ℓ+1 and 1)${}_{\,Y,\ell+1}$
hold.

The above algorithm must terminate after finitely many steps since the
$\mathcal{B}_{X,\ell}$ as defined in (27) form nested sequence of subsets of
the finite set $\mathcal{B}_{X}$ of strictly decreasing cardinality. Indeed,
note that $B_{X}^{\ell+1}$ is chosen from $\mathcal{B}_{X,\ell}$ in the above
algorithm, so that $B_{X}^{\ell+1}\in\mathcal{B}_{X,\ell}$ whilst clearly
$B_{X}^{\ell+1}\notin\mathcal{B}_{X,\ell+1}$. Suppose the algorithm terminates
after the $L$th step. We show that the resulting families
$\mathcal{B}_{X}^{\prime}:=\\{B_{X}^{1},\dots,B_{X}^{L}\\}$ and
$\mathcal{B}_{Y}^{\prime}:=\\{B_{Y}^{1},\dots,B_{Y}^{L}\\}$ satisfy the
desired properties 2) and 3) from the statement of the lemma.

First we note that, provided $R\geq\bar{C}$, Property 3) immediately follows
from Property 3)L of the algorithm.

It remains to show Property 2). By the definition of the stopping condition,
we know either $\mathcal{B}_{X,L}=\emptyset$ or $\mathcal{B}_{Y,L}=\emptyset$.
By symmetry we may assume that $\mathcal{B}_{X,L}=\emptyset$. Using the
standard Vitali covering argument, Property 2)X,L implies that

for all $B_{X}\in\mathcal{B}_{X}$ there exists some $1\leq\ell\leq L$ such
that $B_{X}\subseteq(R/8)\cdot B_{X}^{\ell}$, (34)

provided $R\geq 1$ is chosen sufficiently large depending only on $n$ and
$\lambda$. This is a slightly stronger version of the first inclusion in
Property 2) of the lemma. We turn to the second inclusion. If
$B_{Y}\in\mathcal{B}_{Y}\setminus\mathcal{B}_{Y,L}$, then we may argue as
above to show that $\mathcal{B}_{Y}\subseteq(R/8)\cdot B_{Y}^{\ell}$ for some
$1\leq\ell\leq L$. Thus, it suffices to consider the case
$B_{Y}\in\mathcal{B}_{Y,L}$. By (24) and (34), we know $B_{Y}\cap(R/4)\cdot
B_{X}^{\ell}\neq\emptyset$ for some $1\leq\ell\leq L$. On the other hand,
Property 2)Y,L of the algorithm implies $\mathrm{rad}\,B_{Y}\leq
C\mathrm{rad}\,B_{Y}^{\ell}$ for all $1\leq\ell\leq L$. Thus, (26) and
Property 3)L give us

$B_{Y}\subseteq\bigcup_{\ell=1}^{L}(R/2)\cdot
B_{X}^{\ell}\subseteq\bigcup_{\ell=1}^{L}R\cdot B_{Y}^{\ell},$

again provided $R$ is chosen sufficiently large. ∎

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