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Sbornik: Mathematics, 2025, Volume 216, Issue 3, Pages 368–385
DOI: https://doi.org/10.4213/sm10231e (Mi sm10231) 		 
Sequences of partial sums of multiple trigonometric Fourier series

S. V. Konyaginab

a Steklov Mathematical Institute of Russian Academy of Sciences, Moscow, Russia
b Faculty of Mechanics and Mathematics, Lomonosov Moscow State University, Moscow, Russia
English version PDF (515 kB) HTML full-text Russian version article
References:
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DOI: https://doi.org/10.4213/sm10231e
Abstract: Let $f$ be an integrable $2\pi$-periodic function of $d\ge2$ variables. For a bounded subset $A$ of the $d$-dimensional space let $S_A(f)$ denote the sum of terms of the Fourier series of $f$ with frequencies in $A$. The following problem is addressed: given a sequence $\{A_j\}$ of bounded convex sets, do there exist a function $f$ and a sequence $\{j_\nu\}$ such that $\lim_{\nu\to\infty} |S_{A_{j_\nu}} (f)|=\infty$ almost everywhere?
Bibliography: 5 titles.
Keywords: convergence of multiple trigonometric Fourier series, convex set, lattice.
Funding agency Grant number
Russian Science Foundation 23-71-30001
This work was performed at Lomonosov Moscow State University and supported by the Russian Science Foundation under grant no. 23-71-30001, https://rscf.ru/en/project/23-71-30001/.
Received: 04.11.2024
Published: 16.05.2025
Bibliographic databases:
Document Type: Article
MSC: 42B05, 42B08
Language: English
Original paper language: Russian

§ 1. Introduction

Let $\mathbb T=\mathbb R/2\pi\mathbb Z$ and $d\in\mathbb N$. We denote the usual Euclidean measure on $\mathbb T^d$ by $\operatorname{mes}_d$; the normalized measure is defined by $\mu_d=\operatorname{mes}_d/(2\pi)^d$. Thus, $\mu_d(\mathbb T^d)= 1$. For each $p > 0$, $L^p(\mathbb T^d)$ denotes, as usual, the space of all measurable functions $f\colon \mathbb T^d \to \mathbb C$ satisfying

$$ \begin{equation*} \|f\|_p=\biggl(\int_{\mathbb T^d}|f|^p \,d\mu_d\biggr)^{1/p} < \infty. \end{equation*} \notag $$
This functional defines a norm on $L^p(\mathbb T^d)$ whenever $p \geqslant 1$. For brevity we write $L(\mathbb T^d)=L^1(\mathbb T^d)$.

The trigonometric Fourier series expansion of a function $f\in L(\mathbb T^d)$ is defined by

$$ \begin{equation*} f\sim\sum_{\mathbf k \in \mathbb Z^d} \widehat f(\mathbf k)e^{i\mathbf k\mathbf x}, \qquad \mathbf k= (k_1,\dots,k_d), \quad \mathbf x=(x_1,\dots,x_d), \end{equation*} \notag $$
where
$$ \begin{equation*} \widehat f(\mathbf k)=\int_{\mathbb T^d} f(\mathbf x)e^{-i\mathbf k\mathbf x}\,d\mu_d. \end{equation*} \notag $$

Given a bounded set $A\subset\mathbb R^d$ and a function $f\in L(\mathbb T^d)$, the partial sum of $f$ with respect to $A$ is defined by

$$ \begin{equation*} S_A(f) (\mathbf x)=\sum_{\mathbf k \in \mathbb Z^d\cap A}\widehat f(\mathbf k) e^{i\mathbf k\mathbf x}. \end{equation*} \notag $$

The problem of the convergence of a subsequence of a given sequence of partial sums of a multiple trigonometric Fourier series was studied in [1], where it was shown that the cases of convergence to a finite function and to infinity are substantially different. The present paper is concerned with the latter case.

More precisely, we discuss the following problem. Given a sequence $\{A_j\}$ of bounded subsets of $\mathbb R^d$, find whether there exist a function $f\in L(\mathbb T^d)$ and a sequence $\{j_\nu\}$ such that

$$ \begin{equation*} \lim_{\nu\to\infty} |S_{A_{j_\nu}}(f) (x)|=\infty \end{equation*} \notag $$
almost everywhere. In this case we say that the sequence of sets $\{A_j\}$ satisfies condition $(*)$.

Sufficient conditions for the existence of such a function and a sequence for a fixed system of subsets were found in [1]. Given a bounded set $A\subset\mathbb R^d$, consider the Dirichlet kernel

$$ \begin{equation*} D_A(\mathbf x)=\sum_{\mathbf k\in\mathbb Z^d\cap A} e^{i\mathbf k\mathbf x}. \end{equation*} \notag $$

Theorem 1. Let $\{A_j\}_{j=1}^\infty$ be a sequence of bounded subsets of the space $\mathbb R^d$, let $N_j= |\mathbb Z^d\cap A_j|\geqslant 3$ and

$$ \begin{equation*} \lim_{j\to\infty} \frac{\|D_{A_j}\|_1}{\log N_j}=\infty. \end{equation*} \notag $$
Then the sequence of sets $\{A_j\}$ satisfies condition $(*)$.

It was shown in [1] that Theorem 1 applies to Pringsheim convergence and convergence of partial sums over balls and hyperbolic crosses.

Below we assume that each $A_j$ is convex.

We say that a sequence $\{A_j\}_{j\geqslant1}$ of subsets of $\mathbb R^d$ is absorbing if $\mathbf k\in A_j$ for each vector $\mathbf k\in\mathbb Z^d$ for $j\geqslant j(\mathbf k)$. From Theorem 1 we obtain the following sufficiently general result.

Theorem 2. Let $d\geqslant2$ and $\{A_j\}$ be an absorbing sequence of bounded convex sets in $\mathbb R^d$. Then the sequence $\{A_j\}$ satisfies condition $(*)$.

Corollary 1. Let $d\geqslant2$ and $\{A_j\}$ be a sequence of bounded convex sets in $\mathbb R^d$. Then there exist a function $f\in L(\mathbb T^d)$ and an increasing sequence $\{j_\nu\}_{\nu\geqslant1}$ of natural numbers such that

$$ \begin{equation*} \liminf_{\nu\to\infty} |f(x)-S_{A_{j_\nu}} (f) (x)| > 0 \end{equation*} \notag $$
almost everywhere.

Theorem 2 can be interpreted as saying that any ‘thick’ sequence $\{A_j\}$ of bounded convex sets satisfies condition $(*)$. It can also be conjectured that each ‘thin’ sequence $\{A_j\}$ fails this condition. We show that this conjecture is true.

We start with the one-dimensional case. Let $A_j=[-j,j]$. Then $S_{A_j}(f)$ is the standard $j$th partial sum of the trigonometric Fourier series of the function $f$. From the classical Kolmogorov weak-type inequality for the conjugate function [2] it follows that for any $p\in (0,1)$ and $f\in L(\mathbb T)$ the sequence $\{\|S_{A_j}(f)\|_p\}$ is bounded. As a result, the sequence $\{A_j\}$ does not satisfy condition $(*)$.

In view of this observation one can easily give a multidimensional example of a sequence $\{A_j\}$ failing condition $(*)$. For $d\geqslant2$ let $\mathbf k^{(j)}=(0,\dots,0,-j)$, $\mathbf l^{(j)}=(0,\dots,0,j)$ and $A_j=[\mathbf k^{(j)}, \mathbf l^{(j)}]$. Then the sequence $\{A_j\}$ does not satisfy condition $(*)$.

In this example the sets $A_j$ are ‘one-dimensional’ in the following sense: for each $j$ the set $\mathbb Z^d\cap A_j$ lies on a straight line. We have the following fairly general result on situations in which condition $(*)$ does not hold.

Theorem 3. Let $\{A_j\}_{j=1}^\infty$ be a sequence of bounded convex subsets of $\mathbb R^d$, and let $r\in\mathbb N$ and $0<p<1$. Assume that for each $j$ the set $\mathbb Z^d\cap A_j$ lies on $r$ straight lines. Then for functions $f\in L(\mathbb T^d)$,

$$ \begin{equation*} \sup_j \|S_{A_j}(f)\|_p < \infty. \end{equation*} \notag $$

The following result is a consequence of Theorem 3.

Corollary 2. Under the assumptions of Theorem 3 the sequence of sets $\{A_j\}$ does not satisfy condition $(*)$.

From Theorem 1 and Corollary 2 we can deduce a complete characterization of sequences $\{A_j\}$ of bounded convex subsets of $\mathbb R^2$ that satisfy condition $(*)$.

Theorem 4. Let $d=2$ and $p\in(0,1)$, and let $\{A_j\}$ be a sequence of bounded convex sets in $\mathbb R^d$. Then the following conditions are equivalent:

Some implications in Theorem 4 also hold for each $d>1$. The implications $(1) \to (2) \to (3)$ and $(5) \to (6)$ are clear. The implication $(6) \to (4)$ is the result of Theorem 1, and the implication $(2) \to (5)$ follows from the proof of Theorem 3.

At the same time, many implications in Theorem 4 are no longer true for higher dimensions.

Proposition 1. Let $d > 2$. Then condition (2) in Theorem 4 does not imply condition (1).

Proposition 2. Let $d > 2$. Then condition (3) in Theorem 4 does not imply condition (4).

Proposition 3. Let $d > 2$. Then condition (4) in Theorem 4 does not imply condition (6).

The question of the validity of some implications remains open to this day.

Conjecture 1. Condition (4) in Theorem 4 implies condition (3).

Conjecture 2. Condition (6) in Theorem 4 implies condition (2).

Note that, were Conjecture 2 valid, it would follow from Corollary 2 that condition $(*)$ is equivalent to condition (2) in Theorem 4.

§ 2. Preliminaries

In this section we recall some fundamental concepts from lattice theory. Let ${d\in\mathbb N}$. A lattice in $\mathbb R^d$ is a discrete subgroup $\Lambda$ of the Abelian additive group $\mathbb R^d$. Being discrete means that the distance between any two nonzero elements of $\Lambda$ is bounded away from zero. Another definition is as follows: a lattice is the set of elements of the form $\sum_{j=1}^m k_j \mathbf a^{(j)}$, where $\mathbf a^{(1)}, \dots, \mathbf a^{(m)}$ are linearly independent elements of $\mathbb R^d$ and $k_1,\dots,k_m$ range over the set of integers. Here $m$ is the dimension of the lattice and $(\mathbf a^{(1)}, \dots, \mathbf a^{(m)})$ is a basis of the lattice. The equivalence of these definitions follows from Theorem 7.1 in [3].

A basis of a lattice is not uniquely defined. Moreover, the following result holds (see Proposition 7.1 in [3]).

Lemma 1. Let $\Lambda\subset\mathbb R^d$ be a lattice with basis $(\mathbf a^{(1)}, \dots, \mathbf a^{(d)})$. Then a set of vectors $(\mathbf b^{(1)}, \dots, \mathbf b^{(d)})$ is a basis of the lattice $\Lambda$ if and only if it is the image of $(\mathbf a^{(1)}, \dots, \mathbf a^{(d)})$ under some linear operator with integer matrix in the basis $(\mathbf a^{(1)}, \dots, \mathbf a^{(d)})$ whose determinant is equal to $\pm1$.

Let $\Lambda$ be a lattice in $\mathbb R^d$, and let $l\in\mathbb N$. A system of $l$ vectors in $\Lambda$ is primitive if in is a basis of the $l$-dimensional lattice formed by the intersection of $\Lambda$ with the linear space spanned by these vectors.

Lemma 2. Any primitive system of vectors can be extended to a basis of the lattice $\Lambda$.

This result was actually proved in [3], in the proof of Proposition 7.1.

For the most part, we will consider the lattice $\mathbb Z^d$ of integer vectors (this lattice is important for us because it is the set of frequencies of $d$-dimensional trigonometric series) and sublattices of it.

Let $U$ be an integer matrix of size $d\times d$. Then $U$ maps $\mathbb T^d$ to $\mathbb T^d$. If the determinant of $U$ is $\pm1$, then the inverse matrix $U^{-1}$ exists and is also integer. Hence in this case $U$ is a bijection $\mathbb T^d\to\mathbb T^d$. In addition, $U$ acts locally as the corresponding measure-preserving linear mapping on $\mathbb R^d$, because $\det U=\pm 1$. Hence the bijection $U\colon \mathbb T^d\to\mathbb T^d$ is also measure-preserving.

The next result now follows by a change of the variable.

Lemma 3. Let $U$ be an integer matrix of size $d\times d$ such that $\det U=\pm1$. Then for any function $f\in L(\mathbb T^d)$,

$$ \begin{equation*} \int_{\mathbb T^d} f(U\mathbf x)\,d\mu_d=\int_{\mathbb T^d} f(\mathbf x)\,d\mu_d. \end{equation*} \notag $$

The transpose of a matrix $U$ of size $d\times d$ will be denoted by $U^\top$.

Lemma 4. Let $U$ be an integer matrix of size $d\times d$ such that $\det U=\pm1$. Suppose that $f\in L(\mathbb T^d)$ and $g(\mathbf x)=f(U\mathbf x)$. Then for any $\mathbf k\in\mathbb Z^d$,

$$ \begin{equation*} \widehat g(U^\top\mathbf k)=\widehat f(\mathbf k). \end{equation*} \notag $$

Proof. It can be assumed that the function $f$ is sufficiently smooth, since any integrable function can be approximated by such functions arbitrarily well in $L(\mathbb T^d)$.

The trigonometric Fourier series of a sufficiently smooth function converges to this function uniformly and absolutely. The converse (in a ceratin sense) result also holds: if a trigonometric series converges uniformly to a function, then it is the Fourier series of this function.

So we have the equality

$$ \begin{equation*} f(U\mathbf x)=\sum_{\mathbf k\in\mathbb Z^d} \widehat f(\mathbf k) e^{i\mathbf k(U\mathbf x)} \end{equation*} \notag $$
(with uniform and absolute convergence). Since $\mathbf k(U\mathbf x)=(U^\top\mathbf k)\mathbf x$, this equality can be written as
$$ \begin{equation*} g(\mathbf x)=\sum_{\mathbf k\in\mathbb Z^d} \widehat f(\mathbf k) e^{i(U^\top\mathbf k)\mathbf x}, \end{equation*} \notag $$
which yields the claim of the lemma.

Under the assumptions of Lemma 4 let $A$ be a bounded subset of $\mathbb R^d$. The map $U^\top$ is bijective on $\mathbb Z^d$, and

$$ \begin{equation*} U^\top(A)\cap\mathbb Z^d=U^\top(A\cap\mathbb Z^d). \end{equation*} \notag $$
An application of Lemma 4 shows that
$$ \begin{equation} S_{U^\top(A)}(g)(\mathbf x)=S_A(f)(U\mathbf x). \end{equation} \tag{1} $$
Now the next result follows from this equality and Lemma 3.

Lemma 5. Let $U$ be an integer matrix of size $d\times d$ such that $\det U=\pm1$, let $f\in L(\mathbb T^d)$ and $g(\mathbf x)=f(U\mathbf x)$. Also let $A$ be a bounded subset of $\mathbb R^d$. Then for any ${p > 0}$,

$$ \begin{equation*} \|S_{U^\top(A)}(g)\|_p=\|S_A(f)\|_p. \end{equation*} \notag $$

Lemma 6 will be proved for the $L$-norms of partial sums of Fourier series of trigonometric polynomials (that is, for functions with finitely many nontrivial terms in their Fourier series expansions), although we could have extended the set of norms and the class of functions. Let $d$ and $n$ be natural numbers, $n \leqslant d$. By $\Pi_{d,n}$ (or simply by $\Pi_n$ if $d$ is fixed) we denote the plane

$$ \begin{equation*} \Pi_{d,n}=\{(u_1,\dots,u_d)\in\mathbb R^d\colon u_{n+1}=\dots=u_d=0\} \end{equation*} \notag $$
of dimension $n$.

Lemma 6. For any trigonometric polynomial $f$,

$$ \begin{equation*} \|S_{\Pi_n} (f)\|_1 \leqslant \|f\|_1. \end{equation*} \notag $$

Proof. For $n=d$ the inequality is an obvious equality. Assume that $n < d$.

We need some notation. Given $\mathbf k\in\mathbb Z^d$, we set

$$ \begin{equation*} \mathbf k^-=(k_1,\dots,k_n)\in\mathbb Z^n\quad\text{and} \quad \mathbf k^+=(k_{n+1},\dots,k_d)\in\mathbb Z^{d-n}. \end{equation*} \notag $$
Similarly, for $\mathbf x\in\mathbb T^d$ we set
$$ \begin{equation*} \mathbf x^-=(x_1,\dots,x_n)\in\mathbb T^n\quad\text{and} \quad \mathbf x^+=(x_{n+1},\dots,x_d)\in\mathbb T^{d-n}. \end{equation*} \notag $$
We write $\mathbf k=(\mathbf k^-,0)$ if $\mathbf k\in\mathbb Z^d \cap Pi_n$. Similarly, we write $\mathbf x=(\mathbf x^-,\mathbf x^+)$ if $\mathbf x\in\mathbb T^d$. We denote by $\mu_n^-$ the normalized $n$-dimensional measure on $\mathbb T^n$ relative to the variables $x_1,\dots,x_n$ and by $\mu_{d-n}^+$ the $(d-n)$-dimensional measure on $\mathbb T^{d-n}$ relative to the variables $x_{n+1},\dots,x_d$.

We have

$$ \begin{equation*} S_{\Pi_n} (f)(\mathbf x)=\sum_{\mathbf k^-\in\mathbb Z^n}\widehat f(\mathbf k^-,0) \exp(i\mathbf k^-\mathbf x^-), \end{equation*} \notag $$
where
$$ \begin{equation*} \widehat f(\mathbf k^-,0)=\int_{\mathbb T^n}\biggl(\int_{T^{d-n}} f(\mathbf x)\,d\mu_{d-n}^+\biggr) \exp(-i\mathbf k^-\mathbf x^-) \,d\mu_n^-. \end{equation*} \notag $$
Consider the $n$-variate function
$$ \begin{equation*} g(\mathbf x^-)=\int f(\mathbf x^-,\mathbf x^+)\,d\mu_{d-n}^+. \end{equation*} \notag $$
Then
$$ \begin{equation*} \widehat f(\mathbf k^-,0)=\widehat g(\mathbf k^-)\quad\text{and}\quad S_{\Pi_n} (f)(\mathbf x)=g(\mathbf x^-). \end{equation*} \notag $$
By the definition of $g$,
$$ \begin{equation*} |g(\mathbf x^-)| \leqslant \int_{\mathbb T^{d-n}}|f(\mathbf x^-,\mathbf x^+)|\,d\mu_{d-n}^+. \end{equation*} \notag $$
Integrating this inequality over the set of vectors $\mathbf x^-$ we obtain the required inequality $\|g\|_1 \leqslant \|f\|_1$.

Lemma 6 is proved.

§ 3. Proofs of the implication $(4) \to (3)$ in Theorem 4, of Theorem 2 and Corollary 1

We estimate the Dirichlet kernel of a bounded convex set $A\subset\mathbb R^2$. We can assume that

$$ \begin{equation*} A\cap\mathbb Z^2\neq\varnothing. \end{equation*} \notag $$
We set
$$ \begin{equation*} k_+=\max\{k_2\in\mathbb Z\colon \exists\, k_1\in\mathbb Z\ (k_1,k_2)\in A\} \end{equation*} \notag $$
and
$$ \begin{equation*} k_-=\min\{k_2\in\mathbb Z\colon \exists\, k_1\in\mathbb Z\ (k_1,k_2)\in A\}. \end{equation*} \notag $$
Let $k_2 \in\mathbb Z\cap[k_-,k_+]$,
$$ \begin{equation*} \nu(k_2)=|\{k_1\in\mathbb Z\colon (k_1,k_2)\in A\}|\quad\text{and}\quad \nu=\max_{k_2} \nu(k_2). \end{equation*} \notag $$

Lemma 7. The following inequality holds:

$$ \begin{equation*} \|D_A\|_1 \gg \min((1+\log\nu)^2, (1+\log\nu)\log(2+k_+-k_-)). \end{equation*} \notag $$

The proof depends on some ideas due to Yudin [4].

Proof of Lemma 7. Excluding the trivial cases, we assume that
$$ \begin{equation*} \nu \geqslant 3\quad\text{and} \quad k_+ -k_- \geqslant1. \end{equation*} \notag $$

Let the maximum $\nu$ of the function $\nu(k_2)$ be attained for $k_2=K_2$. One of the numbers $K_2-k_-$ and $k_+-K_2$ is not smaller than $(k_+-k_-)/2$. It can be assumed without loss of generality that

$$ \begin{equation} K_2-k_- \geqslant \frac{k_+-k_-}2. \end{equation} \tag{2} $$
We have
$$ \begin{equation*} \|D_A\|_1=\int_{\mathbb T}\int_{\mathbb T}\biggl|\sum_{k_2 =k_-}^{k_+} \biggl(\sum_{(k_1,k_2)\in A}e^{ik_1 x_1}\biggr) e^{ik_2x_2}\biggr|\, \frac{dx_2}{2\pi}\,\frac{dx_1}{2\pi}. \end{equation*} \notag $$

Integrating with respect to $x_2$ and using the Hardy–Littlewood inequality [5] we obtain

$$ \begin{equation*} \int_{\mathbb T}\biggl|\sum_{k_2 =k_-}^{k_+} \biggl(\sum_{(k_1,k_2)\in A}e^{ik_1 x_1}\biggr) e^{ik_2x_2}\biggr|\, dx_2 \gg \sum_{k_2 =k_-}^{k_+} \frac1{k_2-k_-+1} \biggl|\sum_{(k_1,k_2)\in A}e^{ik_1 x_1}\biggr|. \end{equation*} \notag $$
Next we integrate with respect to $x_1$ (the empty sum is set to be zero). We have
$$ \begin{equation*} \int_{\mathbb T}\biggl|\sum_{(k_1,k_2)\in A}e^{ik_1 x_1}\biggr|\, dx_1 \gg \log(1+\nu(k_2)). \end{equation*} \notag $$
As a result,
$$ \begin{equation} \|D_A\|_1\gg \sum_{k_2 =k_-}^{k_+} \frac1{k_2-k_-+1} \log(1+\nu(k_2)). \end{equation} \tag{3} $$
To prove the lemma, we need to estimate the numbers $\nu(k_2)$ from below, at least for some values of $k_2$.

Along with $\nu(k_2)$, we consider the function

$$ \begin{equation*} \mu(k_2)=\operatorname{mes}(\{x\in\mathbb R\colon (x,k_2)\in A\}). \end{equation*} \notag $$
Here, $\operatorname{mes}=\operatorname{mes}_1$ is the standard one-dimensional Lebesgue measure. The quantities $\nu$ and $\mu$ are related by the inequalities
$$ \begin{equation*} \mu(k_2) -1 \leqslant \nu(k_2) \leqslant \mu(k_2)+1. \end{equation*} \notag $$
In particular, $\nu(K_2) \geqslant \nu-1$. In addition, the function $\mu$ is concave because the set $A$ is convex. Hence
$$ \begin{equation*} \mu(k_2) \geqslant\frac{k_2- k_-}{K_2-k_-} \mu(K_2) \geqslant \frac{k_2- k_-}{K_2-k_-} (\nu-1). \end{equation*} \notag $$
Therefore,
$$ \begin{equation} \nu(k_2) \geqslant \frac{k_2- k_-}{K_2-k_-} (\nu-1)-1. \end{equation} \tag{4} $$
In the rest of the proof of Lemma 7 we use estimates (3) and (4).

First consider the case where

$$ \begin{equation} K_2-k_- \geqslant \nu. \end{equation} \tag{5} $$
Given $j=1,\dots,\nu-2$, we set
$$ \begin{equation*} K_{2,j}=\biggl\{k_2\in\mathbb Z\colon j < \frac{k_2- k_-}{K_2-k_-} (\nu-1) \leqslant j+1 \biggr\}. \end{equation*} \notag $$
Note that each set $K_{2,j}$ is nonempty and $\nu(k_2) \geqslant j$ for $k_2\in K_{2,j}$. By (3),
$$ \begin{equation*} \|D_A\|_1\gg \sum_{j=1}^{\nu-2}(1+\log j)\sum_{k_2\in K_{2,j}} \frac1{k_2-k_-+1}. \end{equation*} \notag $$
From the definition of the sets $K_{2,j}$ we obtain
$$ \begin{equation*} \sum_{k_2\in K_{2,j}} \frac1{k_2-k_-+1} \gg\frac 1j. \end{equation*} \notag $$
Therefore,
$$ \begin{equation*} \|D_A\|_1\gg \sum_{j=1}^{\nu-2}\frac{1+\log j}j \gg (\log\nu)^2. \end{equation*} \notag $$
This proves the claim in the case (5).

Now consider the case where

$$ \begin{equation} \sqrt{\nu-1} \leqslant K_2-k_- \leqslant \nu-1. \end{equation} \tag{6} $$
Excluding the trivial case of small $\nu$, we can assume that $\nu\geqslant10$. In this case $K_2-k_- \geqslant 3$.

From inequality (4) we obtain

$$ \begin{equation*} \nu(k_2)\geqslant k_2-k_--1, \qquad k_- +2 \leqslant k_2 \leqslant K_2. \end{equation*} \notag $$
Substituting these estimates into (3) we find that
$$ \begin{equation*} \|D_A\|_1\gg \sum_{k_2 =k_-+2}^{K_2} \frac1{k_2-k_-+1} \log(k_2-k_-) \gg (\log(K_2-k_-))^2. \end{equation*} \notag $$
In view of (6),
$$ \begin{equation*} \|D_A\|_1\gg (\log\nu)^2. \end{equation*} \notag $$
This proves the claim also in case (6).

Now consider the remaining case when

$$ \begin{equation} K_2-k_- < \sqrt{\nu-1}. \end{equation} \tag{7} $$

As above, we can assume that $\nu\geqslant 10$. From (4) we obtain $\nu(k_2)\gg \nu/(K_2-k_-)$ for all $k_2 =k_- +1,\dots,K_2$. Using (7), we find that $\nu(k_2)\gg \sqrt{\nu}$. Next, an appeal to (3) shows that

$$ \begin{equation*} \|D_A\|_1\gg (\log\sqrt{\nu}) \sum_{k_2 =k_-}^{k_+} \frac1{k_2-k_-+1} \gg(\log\nu)(\log (K_2-k_- +1)). \end{equation*} \notag $$
Recalling (2) we obtain
$$ \begin{equation*} \|D_A\|_1 \gg (\log\nu)(\log (k_+-k_- +2)), \end{equation*} \notag $$
which proves Lemma 7.

Corollary 3. Let $A\subset\mathbb R^2$ be a bounded convex set. Assume that there exists a straight line $l$ such that $\nu=|A \cap \mathbb Z^2 \cap l| \geqslant 2$. Also assume that $A \cap \mathbb Z^2$ contains $\mu \geqslant 1$ points such that no line segment connecting them is parallel to $l$. Then

$$ \begin{equation*} \|D_A\|_1 \gg \min((\log\nu)^2, (\log\nu)\log(1+\mu)). \end{equation*} \notag $$

Proof. There exist integers $k_1$, $k_2$, $u_1$ and $u_2$, $(u_1,u_2)=1$, such that
$$ \begin{equation*} \mathbb Z^2\cap l\cap A=\{(k_1+ku_1,k_2+ku_2)\colon 1\leqslant k\leqslant n\}. \end{equation*} \notag $$
By Lemma 2 the vector $(u_1,u_2)$ can be extended to a basis of the lattice $\mathbb Z^2$. By Lemma 1 this basis is the image of the standard basis $((1,0), (0,1))$ under an integer linear operator $U^\top$ such that $\det U=\det U^\top=\pm1$.

Consider the bounded convex set $\widetilde A= (U^\top)^{-1} A$ and the straight line ${\widetilde l\,{=}\,(U^\top)^{-1} l}$. Note that $U^\top(\mathbb Z^2\cap \widetilde l\cap\widetilde A) =\mathbb Z^2\cap l\cap A$ and the straight line $\widetilde l$ is parallel to the $x$-axis. By the assumption of the corollary the set $\mathbb Z^2\cap \widetilde l\cap\widetilde A$ contains $\nu$ integer points, and the set $\mathbb Z^2\cap\widetilde A$ contains $\mu$ points with different $y$-coordinates.

Set

$$ \begin{equation*} \widetilde f(\mathbf x)=\sum_{\mathbf k \in \widetilde A} e^{i\mathbf k\mathbf x}. \end{equation*} \notag $$
Let $f(\mathbf x)=\widetilde f(U\mathbf x)$. An application of Lemma 4 shows that $\widehat f(\mathbf k)=1$ if $\mathbf k\in A$ and $\widehat f(\mathbf k)=0$ otherwise. So we have
$$ \begin{equation*} f(\mathbf x)=\sum_{\mathbf k \in A} e^{i\mathbf k\mathbf x}. \end{equation*} \notag $$
Hence
$$ \begin{equation*} f=D_A\quad\text{and} \quad \widetilde f=D_{\widetilde A}. \end{equation*} \notag $$
By Lemma 5 $\|D_A\|_1=\|D_{\widetilde A}\|_1$. An appeal to Lemma 7 shows that
$$ \begin{equation*} \|D_{\widetilde A}\|_1 \gg \min((\log\nu)^2, (\log\nu)\log(1+\mu)), \end{equation*} \notag $$
which yields the claim of the corollary.

Lemma 8. Let $d\geqslant2$, let $A\subset\mathbb R^d$ be a bounded convex set, and let $N=|\mathbb Z^d\cap A|\geqslant2$. Then there exists a straight line $l\subset\mathbb R^d$ such that $|\mathbb Z^d\cap l\cap A| \geqslant N^{1/d}$.

Proof. Let $n$ be the greatest natural number such that $n^d < N$. In this case
$$ \begin{equation} (n+1)^d \geqslant N. \end{equation} \tag{8} $$
Let $\mathbb Z_n$ be the set of integers modulo $n$. Consider the canonical map $\phi\colon \mathbb Z^d \to \mathbb Z_n^d$,
$$ \begin{equation*} \phi(\mathbf k)=\phi(k_1,\dots,k_d)=(k_1\ (\operatorname{mod} n),\dots,k_d\ (\operatorname{mod} n)). \end{equation*} \notag $$
Since $|\mathbb Z_n^d|=n^d < N$, there exist $\mathbf k\in \mathbb Z^d\cap A$ and $\mathbf k'\in \mathbb Z^d\cap A$ such that $\phi(\mathbf k)=\phi(\mathbf k')$, that is, $k_j\equiv k'_j\ (\operatorname{mod} n)$ for $j=1,\dots,d$, or $(\mathbf k'-\mathbf k)/n \in \mathbb Z^d$. Let $l$ be the straight line connecting $\mathbf k$ and $\mathbf k'$. The set $\mathbb Z^d\cap l\cap A$ contains $n+1$ points $\mathbf k+(u/n)(\mathbf k'-\mathbf k)$, $u=0,\dots,n$. Now an application of inequality (8) completes the proof.

Lemma 9. Let $A\subset\mathbb R^2$ be a bounded convex set, and let $N=|\mathbb Z^2\cap A|\geqslant2$. Set $\nu=\max|\mathbb Z^2 \cap l \cap A|$, where the maximum is taken over all straight lines $l\subset\mathbb R^2$. Then

$$ \begin{equation*} \|D_A\|_1 \gg (\log N) \biggl(\log\biggl(1+\frac{N}{\nu}\biggr)\biggr). \end{equation*} \notag $$

Proof. Let $l$ be a straight line such that $|\mathbb Z^2 \cap l \cap A|=\nu$. Consider all straight lines parallel to $l$ (including $l$ itself) that contain points in $\mathbb Z^2\cap A$. Each such line contains at most $\nu$ points in $\mathbb Z^2\cap A$. Hence the number of such lines is $\geqslant \mu =[N/\nu]$. We can now use Corollary 3. By Lemma 8 we have $\nu \geqslant N^{1/2}$. Hence
$$ \begin{equation*} \log\nu \asymp \log N\quad\text{and} \quad \log(1+\mu)\asymp \log\biggl(1+\frac N\nu\biggr), \end{equation*} \notag $$
which proves Lemma 9.

The next result, which is a part of Theorem 4, follows from Lemma 9.

Lemma 10. Let $d=2$. Then condition (4) in Theorem 4 implies condition (3).

For a proof of Theorem 2 we need some additional results.

Lemma 11. Let $d\,{\geqslant}\,2$, let $A\,{\subset}\,\mathbb R^d$ be a bounded convex set, and let ${N\,{=}\,|\mathbb Z^d\,{\cap}\, A|\,{\geqslant}\,2}$ and $M\in\mathbb N$. Assume that $A$ contains the cube $[-M,M]^d$. Then there exists a two-dimensional plane $P$ such that the set $\mathbb Z^d\cap P\cap A$ contains $\geqslant N^{1/d}$ points lying on some straight line $l$, and it also contains $M+1$ points lying on a straight line not parallel to $l$.

Proof. By Lemma 8 there exists a straight line $l$ containing $\geqslant N^{1/d}$ points in $\mathbb Z^d\cap A$. Let $\mathbf x$ be one such point. Let $\mathbf y\in[1,M]^d\cap\mathbb Z^d$ be a point such that $x_j\equiv y_j \ (\operatorname{mod} M)$, $j=1,\dots,N$. We have $(\mathbf y-\mathbf x)/M \in \mathbb Z^d$.

If $\mathbf y\notin l$, then we have $M+1$ points

$$ \begin{equation*} \mathbf x+\frac uM(\mathbf y-\mathbf x), \qquad u=0,\dots,M. \end{equation*} \notag $$
All these points lie in $\mathbb Z^d\cap A$ and also on the straight line connecting $\mathbf x$ and $\mathbf y$, which proves the claim.

Let $\mathbf y\in l$. If the vector $(1,0,\dots,0)$ is not parallel to $l$, then we set $\mathbf y'=\mathbf y-(M,0,\dots,0)\notin l$. The straight line connecting $\mathbf x$ and $\mathbf y'$ (which is not parallel to $l$) contains the following $M+1$ points:

$$ \begin{equation*} \mathbf x+\frac uM(\mathbf y'-\mathbf x)\in\mathbb Z^d\cap A, \qquad u=0,\dots,M. \end{equation*} \notag $$

Assume now that $\mathbf y\in l$ and the vector $(1,0,\dots,0)$ is parallel to $l$. The vector $(0,1,\dots,0)$ is not parallel to $l$, and so we obtain a straight line connecting the points $\mathbf x$ and $\mathbf y''=\mathbf y-(0,M,\dots,0)\notin l$.

Lemma 11 is proved.

Lemma 12. Under the assumptions of Lemma 11,

$$ \begin{equation*} \|D_A\|_1 \gg (\log N) (\log(1+M)). \end{equation*} \notag $$

Proof. We proceed as in the proof of Corollary 3; some identical steps will be omitted.

There exist integer vectors $\mathbf u,\mathbf v\in\mathbb Z^d$ such that

$$ \begin{equation*} \mathbb Z^d\cap l\cap A=\{\mathbf v+k\mathbf u\colon 1\leqslant k\leqslant K\}, \qquad K \geqslant N^{1/d}, \end{equation*} \notag $$
and the coordinates of the vector $\mathbf u$ have no common divisor. By Lemma 2, the set $\{\mathbf u\}$ can be extended to a basis of the lattice $\mathbb Z^d\cap P'$, where $P'\subset\mathbb R^d$ is the two-dimensional subspace parallel to the plane $P$. Another appeal to Lemma 2 shows that the above two-element basis of the lattice $\mathbb Z^d\cap P'$ can be extended to a basis of $\mathbb Z^d$. An application of Lemma 1 shows that this latter is the image of the standard basis $((1,0,\dots,0),(0,1,0,\dots,0),\dots,(0,\dots,1))$ under an integer linear operator $U^\top$ such that $\det U=\det U^\top=\pm1$.

Note that $(U^\top)^{-1} l =\widetilde l$ is a straight line parallel to $(1,0,\dots,0)$, and ${(U^\top)^{-1} P \,{=}\,\widetilde P}$ is a two-dimensional plane parallel to the space spanned by the first two vectors $(1,0,\dots,0)$ and $(0,1,0,\dots,0)$ of the standard basis of $\mathbb Z^d$. We also have $\widetilde l \subset \widetilde P$. The set $\widetilde A=(U^\top)^{-1} A$ is convex and bounded, $\mathbb Z^d\cap \widetilde l \cap\widetilde A$ contains $\geqslant N^{1/d}$ points, and the set $\mathbb Z^d\cap \widetilde P \cap \widetilde A$ contains $M+1$ points lying on a straight line not parallel to $\widetilde l$.

In the proof of Corollary 3 it was shown that

$$ \begin{equation} \|D_{\widetilde A}\|_1=\|D_A\|_1. \end{equation} \tag{9} $$
So it suffices to estimate $\|D_{\widetilde A}\|_1$ from below.

Given an arbitrary point $\mathbf k^0\in\mathbb Z^d\cap \widetilde l \cap\widetilde A$, we set $A_0=\widetilde A-\mathbf k^0$, $P_0=\widetilde P-\mathbf k^0$ and $l_0=\widetilde l-\mathbf k^0$. Note that, in the notation of Lemma 6,

$$ \begin{equation*} l_0=\Pi_1\quad\text{and} \quad P_0=\Pi_2. \end{equation*} \notag $$
It is easily seen that
$$ \begin{equation} \|D_{A_0}\|_1=\|D_{\widetilde A}\|_1. \end{equation} \tag{10} $$
By Lemma 6
$$ \begin{equation} \|D_{A_0}\|_1 \geqslant \|D_{A_0\cap P_0}\|_1. \end{equation} \tag{11} $$
An appeal to Corollary 3 shows that
$$ \begin{equation} \|D_{A_0\cap P_0}\|_1 \gg (\log N) (\log(1+M)). \end{equation} \tag{12} $$
Now a combination of equalities and estimates (9)–(12) completes the proof.

Lemma 12 is proved.

Proof of Theorem 2. The assumption that the system $\{A_j\}$ is absorbing implies that for each $M\in\mathbb N$ and for all sufficiently large $j$ the set $A_j$ contains the cube $[-M,M]^d$. From Lemma 12 we obtain
$$ \begin{equation*} \lim_{j\to\infty} \frac{\|D_{A_j}\|_1}{\log |\mathbb Z^d\cap A_j|}=\infty. \end{equation*} \notag $$
Now an application of Theorem 1 completes the proof.
Proof of Corollary 1. If the system $\{A_j\}$ is absorbing, then the required result is secured by Theorem 2. Otherwise there exist $\mathbf k\in\mathbb Z^d$ and a subsequence $\{j_\nu\}_{\nu\geqslant1}$ such that $\mathbf k\notin A_{j_\nu}$. In this case it suffices to put $f(\mathbf x)=\exp(i\mathbf k\mathbf x)$, completing the proof.

§ 4. Proof of Theorem 3

Proof of Theorem 3. Our argument depends on estimates for Fourier sums of one-dimensional functions. Given $f\in L(\mathbb T)$, the Riesz operator is defined by
$$ \begin{equation*} Rf(x)=\sum_{k\geqslant0} \widehat f(k) e^{ikx}. \end{equation*} \notag $$
The operator $R$ is well defined for any $f\in L^2(\mathbb T)$ since the series is convergent in $L^2(\mathbb T)$. This series can naturally be extended to the whole space $L(\mathbb T)$ (see the comment below).

By Kolmogorov’s theorem [2] mentioned above, for each $p\in(0,1)$ there exists a positive number $C'(p) $ such that, for any function $f\in L(\mathbb T)$,

$$ \begin{equation} \|R(f)\|_p \leqslant C'(p)\|f\|_1. \end{equation} \tag{13} $$

Inequality (13) can first be verified for $f\in L^2(\mathbb T)$, and then, using the density of the space $L^2(\mathbb T)$ in $L(\mathbb T)$ and the continuity of $R$, as an operator from $L(\mathbb T)$ to $L^p(\mathbb T)$, on $L^2(\mathbb T)$, we can extend this operator to the whole of $L(\mathbb T)$ with the preservation of inequality (13).

Given $f\in L(\mathbb T)$ and $m\leqslant n$, consider the partial sums

$$ \begin{equation*} S_{m,n} (f)(x)=\sum_{k =m}^n \widehat f(k)e^{ikx}. \end{equation*} \notag $$
A similar inequality for $ S_{m,n}(f)$ can easily be obtained from (13).

Lemma 13. For any $p\in(0,1)$ there exists a number $C(p)$ such that, for any function $f\in L(\mathbb T)$ and all integers $m\leqslant n$,

$$ \begin{equation*} \|S_{m,n}(f)\|_p \leqslant C(p)\|f\|_1. \end{equation*} \notag $$

Proof. Setting
$$ \begin{equation*} f_1(x)=e^{-imx} f(x)\quad\text{and} \quad f_2(x)=e^{-i(n+1)x} f(x), \end{equation*} \notag $$
we have
$$ \begin{equation} S_{m,n}(f)(x)= Rf_1(x) e^{imx}-Rf_2(x) e^{i(n+1)x}. \end{equation} \tag{14} $$
For $f\in L^2(\mathbb T)$ equality (14) is verified by expanding the corresponding functions in trigonometric series, and for an arbitrary function $f\in L(\mathbb T)$ the result is obtained by taking the limit.

Now the conclusion of the lemma is secured by inequality (13) and equality (14). Here we can take $ C(p)=2^{1/p}C'(p)$.

The next result, which generalizes Lemma 13 to one-dimensional sums of multiple Fourier series, is a key ingredient of the proof of Theorem 3.

Lemma 14. Let $p\in(0,1)$ be arbitrary. Then there exists a number $C(p)$ such that

$$ \begin{equation} \|S_A( f)\|_p \leqslant C(p)\|f\|_1 \end{equation} \tag{15} $$
for any function $f\in L(\mathbb T^d)$ and any bounded convex set $A\subset \mathbb R^d$ such that the set $\mathbb Z^d\cap A$ lies on a straight line.

Note that $C(p)$ is the same as in Lemma 13.

Proof of Lemma 14. 1. First consider the case where $\mathbb Z^d\cap A$ lies on the straight line $\{\mathbf u =(u_1,\dots,u_d)\colon u_1 =\dots=u_{d-1}=0\}$. Let
$$ \begin{equation*} \mathbb Z^d\cap A=\{ (0,\dots,0,k)\colon m\leqslant k \leqslant n\}. \end{equation*} \notag $$
Consider the one-dimensional function
$$ \begin{equation*} f_1(x)=\int_{\mathbb T^{d-1}} f(0,\dots,0,x)\, d\mu_{d-1}. \end{equation*} \notag $$
It is easily seen that $f_1\in L(\mathbb T)$ and that
$$ \begin{equation} \|f_1\|_1 \leqslant \|f\|_1. \end{equation} \tag{16} $$
Next we have
$$ \begin{equation*} \widehat f(0,\dots,k)=\widehat f_1 (k), \end{equation*} \notag $$
hence
$$ \begin{equation*} S_A (f)(x_1,\dots,x_d)=S_{m,n}(f_1)(x_d)\quad\text{and}\quad \|S_A (f)\|_p=\|S_{m,n}(f_1)\|_p. \end{equation*} \notag $$
Now inequality (15) follows from the last equality and (16).

2. Assume now that $\mathbb Z^d\cap A$ lies on a straight line $\{\mathbf u =(u_1,\dots,u_d)\colon u_1 =k_1, \dots, u_{d-1}=k_{d-1}\}$. Consider the function

$$ \begin{equation*} f_1(\mathbf x)=f(\mathbf x)e^{-i\widetilde{\boldsymbol k}\mathbf x}, \end{equation*} \notag $$
where $\widetilde{\boldsymbol k}=(k_1,\dots,k_{d-1},0)$. We have
$$ \begin{equation*} \widehat f(\mathbf k)=\widehat f_1(\mathbf k'), \qquad \mathbf k'=\mathbf k-\widetilde{\boldsymbol k}. \end{equation*} \notag $$
Setting $A'=A-\widetilde{\boldsymbol k}$ we find that
$$ \begin{equation*} S_A(f)=S_{A'}(f_1) t^{i\widetilde{\boldsymbol k}\mathbf x}. \end{equation*} \notag $$
For the set $A'$ we have the case already considered, and so
$$ \begin{equation*} \|S_{A'}( f_1)\|_p \leqslant C(p)\|f_1\|_1, \end{equation*} \notag $$
which proves the required inequality (15).

3. We can now prove Lemma 14 in the general case. If $\mathbb Z^d\cap A$ is a singleton, there is nothing to prove. Assume that this set contains at least two points and lies on a straight line $l$.

Let $\mathbf b$ be a vector with coprime coordinates parallel to $l$. By Lemma 2 $\mathbf b$ can be extended to a basis of $\mathbb Z^d$. It will be convenient to order this basis so that $\mathbf b$ is the last vector in it. By Lemma 2 this basis is the image of the canonical basis $(\mathbf a^{(1)}, \dots, \mathbf a^{(d)})$ of $\mathbb Z^d$ (that is, $a_i^j=1$ for $i=j$ and $a_i^j=0$ for $i\neq j$) under some linear operator whose matrix $U^\top$ has integer entries and determinant $\pm1$. In particular,

$$ \begin{equation} U^\top(\mathbf a^d) =\mathbf b. \end{equation} \tag{17} $$

Consider the closed bounded set $A_1=(U^\top)^{-1} A \subset \mathbb R^d$. By equality (17), $\mathbb Z^d\cap A_1$ lies on some straight line $\{\mathbf u =(u_1,\dots,u_d)\colon u_1 =k_1, \dots, u_{d-1}=k_{d-1}\}$. The matrix $I$ also has integer entries and determinant $\pm1$. Consider the function ${f_1(\mathbf x)=f(U^{-1}\mathbf x)}$. By Lemmas 3 and 5,

$$ \begin{equation*} \|f\|_1=\|f_1\|_1\quad\text{and} \quad\|S_{A}(f)\|_p=\|S_{A_1}(f_1)\|_p. \end{equation*} \notag $$
For the set $A_1$ we have case 2, and so
$$ \begin{equation*} \|S_{A_1}( f_1)\|_p \leqslant C(p)\|f_1\|_1, \end{equation*} \notag $$
which yields the required inequality (15).

Lemma 14 is proved.

For the proof of Theorem 3 we need the following generalization of Lemma 14.

Lemma 15. Let $p\in(0,1)$ and $r\in\mathbb N$ be arbitrary. Then there exists a number $C(p,r)$ such that

$$ \begin{equation*} \|S_A( f)\|_p \leqslant C(p,r)\|f\|_1 \end{equation*} \notag $$
for any function $f\in L(\mathbb T^d)$ and any bounded convex set $A\subset\mathbb R^d$ such that the set $\mathbb Z^d\cap A$ lies on $r$ straight lines.

Proof. Let
$$ \begin{equation*} \mathbb Z^d\cap A \subset \bigcup_{j=1}^r l_j, \end{equation*} \notag $$
where $l_1,\dots,l_r$ are different straight lines. Two lines $l_i$ and $l_j$ have at most one common point, and so
$$ \begin{equation} S_A(f)=\sum_{j=1}^r S_{A\cap l_j}(f) - \sum_{j=1}^R S_{B_j} (f), \end{equation} \tag{18} $$
where $R\leqslant r(r-1)/2$ and $B_1,\dots,B_R$ are (non necessarily different) one-point subsets of $\mathbb Z^d$.

By Lemma 14, for $j=1,\dots,r$ we have

$$ \begin{equation} \|S_{A_j}( f)\|_p \leqslant C(p)\|f\|_1. \end{equation} \tag{19} $$
Next, let $j=1,\dots,R$ and $B_j=\{\mathbf k\}\subset\mathbb Z^d$. Then
$$ \begin{equation} |\|S_{B_j}( f)\|_p|=|\widehat f(\mathbf k)| \leqslant \|f\|_1. \end{equation} \tag{20} $$
Now the conclusion of the lemma follows from equality (18) and inequalities (19) and (20).

Lemma 15 is proved.

Theorem 3 is immediate from Lemma 15.

§ 5. Proof of Theorem 4

Lemma 16. Let $d=2$, let $A\subset\mathbb R^2$ be a bounded convex set, $|\mathbb Z^2\cap A|=N > 0$, let $0 < c \leqslant 1$, and let $|\mathbb Z^2\cap l\cap A|\geqslant cN$ for some straight line $l\subset\mathbb R^2$. Then the set $\mathbb Z^2\cap A$ lies on fewer than $9/c$ straight lines parallel to $l$.

Proof. If $|\mathbb Z^2\cap l\cap A|=1$, then $c \leqslant 1/N$, and the required result is clear. So we assume that
$$ \begin{equation} |\mathbb Z^2\cap l\cap A| \geqslant 2. \end{equation} \tag{21} $$

1. First consider the particular case where $l=\{(x,0)\colon x\in\mathbb R\}$ and

$$ \begin{equation*} \mathbb Z^2\cap l\cap A=\{(k,0)\colon 1\leqslant k\leqslant n\}. \end{equation*} \notag $$
Let $m$ be such that
$$ \begin{equation*} |m|=\max\{|k_2|\colon \exists\, k_1\mid (k_1,k_2)\in \mathbb Z^2\cap A\} \end{equation*} \notag $$
and $(k_1,m) \in A$ for some $k_1\in\mathbb Z$. Our aim is to estimate $|m|$ from above.

Consider the triangle $P$ with vertices at the points $(1,0)$, $(n,0)$ and $(k_1,m)$. Note that $P\subset A$. By (21) we have $n\geqslant2$. The area of the triangle $P$ has the estimate $ S=(n-1)|m|/2 \geqslant cN|m|/4$. By Pick’s theorem the number of integer points in $P$ and on its boundary is greater than $S$. Hence $ N > cN|m|/4$, that is, $|m| < 4/c$.

Thus, the set $\mathbb Z^2\cap B$ lies on fewer than $1+8/c \leqslant 9/c$ parallel straight lines $l_k=\{(x_1,k)\colon |k| \leqslant |m|\}$.

2. Let the straight line $l$ be parallel to the $x$-axis. Then there exist $k_1,k_2\in\mathbb Z$ and $n\in\mathbb N$ such that

$$ \begin{equation*} \mathbb Z^2\cap l\cap A=\{(k_1+k,k_2)\colon 1\leqslant k\leqslant n\}. \end{equation*} \notag $$
Consider the set $\widetilde A =A-(k_1,k_2)$. We have $\mathbb Z^2\cap l\cap \widetilde A=\{(k,0)\colon 1\leqslant k\leqslant n\}$. By the above $\mathbb Z^2\cap \widetilde A$ lies on fewer than $9/c$ straight lines parallel to $l$. The set $\mathbb Z^2\cap A$ also lies on fewer than $9/c$ straight lines parallel to $l$, as required.

3. Now consider the general case. In this case there exist integers $k_1$, $k_2$, $u_1$ and $u_2$, $(u_1,u_2)=1$, such that

$$ \begin{equation*} \mathbb Z^2\cap l\cap A=\{(k_1+ku_1,k_2+ku_2)\colon 1\leqslant k\leqslant n\}. \end{equation*} \notag $$
By Lemma 2 the vector $(u_1,u_2)$ can be extended to a basis of $\mathbb Z^2$. Lemma 1 shows that this basis is the image of the standard basis $((1,0), (0,1))$ under and integer linear operator $U$ such that $\det U=\pm1$. Consider the set $\widetilde A=U^{-1}A$ and the straight line $\widetilde l=U^{-1} l$. We have $U(\mathbb Z^2\cap \widetilde l\cap\widetilde A) =\mathbb Z^2\cap l\cap A$ and $\widetilde l$ is parallel to the $x$-axis. By the above the set $\mathbb Z^2\cap \widetilde A$ lies on fewer than $9/c$ straight lines parallel to $\widetilde l$. Hence $\mathbb Z^2\cap A$ can also be covered by fewer than $9/c$ straight lines parallel to $l$.

Lemma 16 is proved.

Now we are ready to prove Theorem 4.

Proof of Theorem 4. Consider the chain of implications
$$ \begin{equation} (1) \to (2) \to (5) \to (6) \to (4) \to (3) \to (1). \end{equation} \tag{22} $$
The implications $(1) \to (2)$ and $(5) \to (6)$ are clear. The implication $(2) \to (5)$ is Theorem 3, and $(6) \to (4)$ is Theorem 1. The implication $(4) \to (3)$ is Lemma 10, and $(3) \to (1)$ is immediate from Lemma 16.

The chain of implications (22) shows that for $d=2$ conditions (1)–(6) are equivalent. This proves Theorem 4.

§ 6. Proofs of Propositions 1–3

Note that it suffices to prove all propositions in the three-dimensional case, because the examples constructed for $d=3$ can be carried over to spaces of higher dimension.

The construction of examples for $d=3$ is based on the following idea.

Let $B_0$ be a nonempty convex subset of the hyperplane

$$ \begin{equation*} H_0=\{(x_1,x_2,0)\in \mathbb R^3\}, \end{equation*} \notag $$
and $B_0$ be a nonempty convex subset of the hyperplane
$$ \begin{equation*} H_1=\{(x_1,x_2,1)\in \mathbb R^3\}; \end{equation*} \notag $$
let $B$ be the convex hull of the set $B_0\cup B_1$. Then $B$ consists of the points of the form $\lambda \mathbf x+(1-\lambda) \mathbf y$, where $\mathbf x \in B_0$, $\mathbf y \in B_1$ and $\lambda\in[0,1]$. The third coordinate of this linear combination is equal to $1-\lambda$. This is an integer if and only if $\lambda=0$ or $\lambda=1$. Hence
$$ \begin{equation*} \mathbb Z^3\cap B=(\mathbb Z^3\cap B_0)\cup(\mathbb Z^3\cap B_1). \end{equation*} \notag $$

Proof of Proposition 1. For $j\geqslant1$ set
$$ \begin{equation*} \begin{gathered} \, A_{0,j}=\{(x_1, 0, 0)\colon 0 \leqslant x_1\leqslant j\}, \\ A_{1,j}=\{(0, x_2, 1)\colon 0 \leqslant x_2\leqslant j\} \end{gathered} \end{equation*} \notag $$
and
$$ \begin{equation*} A_j=\operatorname{conv}(A_{0,j} \cup A_{1,j}). \end{equation*} \notag $$
The set $A_j$ can be covered by two straight lines. At the same time, for any straight line $l$ either each straight line parallel to $l$ has at most one common point with $A_{0,j}$ or each straight line parallel to $l$ has at most one common point with $A_{1,j}$. Hence condition (1) is not met.

This proves the proposition.

Proof of Proposition 2. For $j\geqslant1$ set
$$ \begin{equation*} \begin{gathered} \, A_{0,j}=\{(x_1, 0, 0)\colon 0 \leqslant x_1\leqslant j\}, \\ A_{1,j}=\{(x_1, x_2, 1)\colon 0 \leqslant x_1,x_2\leqslant \sqrt j\} \end{gathered} \end{equation*} \notag $$
and
$$ \begin{equation*} A_j=\operatorname{conv}(A_{0,j} \cup A_{1,j}). \end{equation*} \notag $$
Then the straight line $(x_1,0,0)$ contains at least half the set $\mathbb Z^3\cap A$, and so condition (3) is fulfilled. At the same time, for $j\geqslant2$ we have
$$ \begin{equation*} \biggl\|\sum_{\mathbf k\in \mathbb Z^3\cap A_{0,}j} \exp(i\mathbf k\mathbf x)\biggr\|_1 \ll \log j \end{equation*} \notag $$
and
$$ \begin{equation*} \biggl\|\sum_{\mathbf k\in \mathbb Z^3\cap A_{1,}j} \exp(i\mathbf k\mathbf x)\biggr\|_1 \gg (\log j)^2. \end{equation*} \notag $$
Therefore,
$$ \begin{equation*} \biggl\|\sum_{\mathbf k\in \mathbb Z^3\cap A_{}j} \exp(i\mathbf k\mathbf x)\biggr\|_1 \gg (\log j)^2, \end{equation*} \notag $$
and so condition (4) is not met.

This proves the proposition.

Proof of Proposition 3. For $j\geqslant1$ we set
$$ \begin{equation*} \begin{gathered} \, A_{0,j}=\{(x_1, 0, 0)\colon -j \leqslant x_1\leqslant j\}, \\ A_{1,j}=\{(x_1, x_2, 1)\colon 0 \leqslant x_1,x_2\leqslant 1+\log j\} \end{gathered} \end{equation*} \notag $$
and
$$ \begin{equation*} A_j=\operatorname{conv}(A_{0,j} \cup A_{1,j}). \end{equation*} \notag $$
Then for $j\geqslant2$ we have
$$ \begin{equation*} \biggl\|\sum_{\mathbf k\in \mathbb Z^3\cap A_{0,}j} \exp(i\mathbf k\mathbf x)\biggr\|_1 \ll \log j \end{equation*} \notag $$
and
$$ \begin{equation} \biggl\|\sum_{\mathbf k\in \mathbb Z^3\cap A_{1,}j} \exp(i\mathbf k\mathbf x)\biggr\|_1 \asymp (1+\log\log j)^2. \end{equation} \tag{23} $$
Therefore,
$$ \begin{equation*} \biggl\|\sum_{\mathbf k\in \mathbb Z^3\cap A_{}j} \exp(i\mathbf k\mathbf x)\biggr\|_1 \ll \log j, \end{equation*} \notag $$
and so condition (4) is met.

It remains to show that the sequence $\{A_j\}$ satisfies condition $(*)$.

Consider an arbitrary function $f\in L(\mathbb T^3)$ and the one-dimensional function $g\in L(\mathbb T)$ defined by

$$ \begin{equation*} g(x_1)=\int_{-\pi}^\pi \int_{-\pi}^\pi f(x_1,x_2,x_3)\, \frac{dx_2}{2\pi}\,\frac{dx_3}{2\pi}. \end{equation*} \notag $$
It is easily seen that
$$ \begin{equation*} \widehat g(k)=\widehat f(k,0,0), \end{equation*} \notag $$
and therefore
$$ \begin{equation} S_j(g)(x_1)=S_{A_{0,j}}(x_1,x_2,x_3). \end{equation} \tag{24} $$

From Theorem 1 and estimate (23) it follows that there exist a function ${f\,{\in}\, L(\mathbb T^3)}$ and an increasing sequence $\{j_\nu\}$ such that

$$ \begin{equation*} \lim_{\nu\to\infty} |S_{A_{1,j_\nu}}(f) (x)|=\infty \end{equation*} \notag $$
almost everywhere. Next, by [2] the sequence $\{j_\nu\}$ contains a subsequence $\{j'_\nu\}$ such that $S_{j'_\nu}(g)(x_1)$ converges to $g(x_1)$ almost everywhere. From (24) we see that
$$ \begin{equation*} \lim_{\nu\to\infty} S_{A_{1,j'_\nu}}(f)(x)=g(x_1) \end{equation*} \notag $$
almost everywhere on $\mathbb T^3$. Hence it follows from the previous equality that
$$ \begin{equation*} \lim_{\nu\to\infty} |S_{A_{j'_\nu}}(f) (x)|=\infty \end{equation*} \notag $$
almost everywhere, as required. This proves Proposition 3.

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1. S. V. Konyagin, “On Pringsheim convergence of a subsequence of partial sums of a multiple trigonometric Fourier series”, Proc. Steklov Inst. Math., 323 (2023), 159–172  mathnet  crossref  mathscinet  zmath
2. A. Kolmogoroff (Kolmogorov), “Sur les fonctions harmoniques conjuguées et les séries de Fourier”, Fund. Math., 7 (1925), 24–29  crossref  zmath
3. O. N. German and Yu. V. Nesterenko, Number theoretic methods in cryptography, Akademia Publishing Centre, Moscow, 2012, 270 pp. (Russian)  zmath
4. V. A. Yudin, “A lower bound for Lebesgue constants”, Math. Notes, 25:1 (1979), 63–65  mathnet  crossref  mathscinet  zmath
5. G. H. Hardy and J. E. Littlewood, “Some new properties of Fourier constants”, Math. Ann., 97:1 (1927), 159–209  crossref  mathscinet  zmath
Citation: S. V. Konyagin, “Sequences of partial sums of multiple trigonometric Fourier series”, Sb. Math., 216:3 (2025), 368–385
Citation in format AMSBIB
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\by S.~V.~Konyagin
\paper Sequences of partial sums of multiple trigonometric Fourier series
\jour Sb. Math.
\yr 2025
\vol 216
\issue 3
\pages 368--385
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\crossref{https://doi.org/10.4213/sm10231e}
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