arXiv subjects

Dmitrii Zhelezov

Publications and source records attributed to Dmitrii Zhelezov.

13 recordsLinked to original sources

arXiv2026

The sum-product conjecture is false for real numbers

We disprove the sum-product conjecture for real numbers by constructing arbitrarily large $A\subset \mathbb{R}$ (whose elements are algebraic integers in a number field of degree $\asymp \log\lvert A\rvert$) such that \[\max(\lvert A+A\rvert ,\lvert AA\rvert)\leq \lvert A\rvert^{2-c}\] where $c>0$ is an absolute constant. We also disprove the many sums and products conjecture by constructing, for any $k\geq 3$, arbitrarily large $A\subset \mathbb{R}$ such that \[\max(\lvert kA\rvert,\lvert A^{(k)}\rvert)\leq \lvert A\rvert^{C\frac{\log k}{\log\log k}}\] for some constant $C>0$. We obtain similar constructions for $p$-adics, finite fields, and function fields in positive characteristic, and also obtain new lower bounds for the number of solutions to linear equations in a multiplicative group and the number of solutions to the unit equation in sufficiently many variables.

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arXiv2023

Convexity, Elementary Methods, and Distances

This paper considers an extremal version of the Erdős distinct distances problem. For a point set $P \subset \mathbb R^d$, let $Δ(P)$ denote the set of all Euclidean distances determined by $P$. Our main result is the following: if $Δ(A^d) \ll |A|^2$ and $d \geq 5$, then there exists $A' \subset A$ with $|A'| \geq |A|/2$ such that $|A'-A'| \ll |A| \log |A|$. This is one part of a more general result, which says that, if the growth of $|Δ(A^d)|$ is restricted, it must be the case that $A$ has some additive structure. More specifically, for any two integers $k,n$, we have the following information: if \[ | Δ(A^{2k+3})| \leq |A|^n \] then there exists $A' \subset A$ with $|A'| \geq |A|/2$ and \[ | kA'- kA'| \leq k^2|A|^{2n-3}\log|A|. \] These results are higher dimensional analogues of a result of Hanson, who considered the two-dimensional case.

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arXiv2023

The sum-product problem for integers with few prime factors

It was asked by E. Szemerédi if, for a finite set $A\subset\mathbb{Z}$, one can improve estimates for $\max\{|A+A|,|A\cdot A|\}$, under the constraint that all integers involved have a bounded number of prime factors -- that is, each $a\in A$ satisfies $ω(a)\leq k$. In this paper, answer Szemerédi's question in the affirmative by showing that this maximum is of order $|A|^{\frac{5}{3}-o(1)}$ provided $k\leq (\log|A|)^{1-ε}$ for some $ε>0$. In fact, this will follow from an estimate for additive energy which is best possible up to factors of size $|A|^{o(1)}$.

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arXiv2022

Query complexity and the polynomial Freiman-Ruzsa conjecture

We prove a query complexity variant of the weak polynomial Freiman-Ruzsa conjecture in the following form. For any $ε> 0$, a set $A \subset \mathbb{Z}^d$ with doubling $K$ has a subset of size at least $K^{-\frac{4}ε}|A|$ with coordinate query complexity at most $ε\log_2 |A|$. We apply this structural result to give a simple proof of the "few products, many sums" phenomenon for integer sets. The resulting bounds are explicit and improve on the seminal result of Bourgain and Chang.

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arXiv2020

On iterated product sets with shifts II

The main result of this paper is the following: for all $b \in \mathbb Z$ there exists $k=k(b)$ such that \[ \max \{ |A^{(k)}|, |(A+u)^{(k)}| \} \geq |A|^b, \] for any finite $A \subset \mathbb Q$ and any non-zero $u \in \mathbb Q$. Here, $|A^{(k)}|$ denotes the $k$-fold product set $\{a_1\cdots a_k : a_1, \dots, a_k \in A \}$. Furthermore, our method of proof also gives the following $l_{\infty}$ sum-product estimate. For all $γ>0$ there exists a constant $C=C(γ)$ such that for any $A \subset \mathbb Q$ with $|AA| \leq K|A|$ and any $c_1,c_2 \in \mathbb Q \setminus \{0\}$, there are at most $K^C|A|^γ$ solutions to \[ c_1x + c_2y =1 ,\,\,\,\,\,\,\, (x,y) \in A \times A. \] In particular, this result gives a strong bound when $K=|A|^ε$, provided that $ε>0$ is sufficiently small, and thus improves on previous bounds obtained via the Subspace Theorem. In further applications we give a partial structure theorem for point sets which determine many incidences and prove that sum sets grow arbitrarily large by taking sufficiently many products. We utilise a query-complexity analogue of the polynomial Freiman-Ruzsa conjecture, due to Zhelezov and Pálvölgyi. This new tool replaces the role of the complicated setup of Bourgain and Chang, which we had previously used. Furthermore, there is a better quantitative dependence between the parameters.

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arXiv2020

An analytic approach to cardinalities of sumsets

Let $d$ be a positive integer and $U \subset \mathbb{Z}^d$ finite. We study $$β(U) : = \inf_{\substack{A , B \neq \emptyset \\ \text{finite}}} \frac{|A+B+U|}{|A|^{1/2}{|B|^{1/2}}},$$ and other related quantities. We employ tensorization, which is not available for the doubling constant, $|U+U|/|U|$. For instance, we show $$β(U) = |U|,$$ whenever $U$ is a subset of $\{0,1\}^d$. Our methods parallel those used for the Prékopa-Leindler inequality, an integral variant of the Brunn-Minkowski inequality.

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arXiv2020

A Weighted Prékopa-Leindler inequality and sumsets with quasicubes

We give a short, self-contained proof of two key results from a paper of four of the authors. The first is a kind of weighted discrete Prékopa-Leindler inequality. This is then applied to show that if $A, B \subseteq \mathbb{Z}^d$ are finite sets and $U$ is a subset of a "quasicube" then $|A + B + U| \geq |A|^{1/2} |B|^{1/2} |U|$. This result is a key ingredient in forthcoming work of the fifth author and Pälvölgyi on the sum-product phenomenon.

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arXiv2018

On iterated product sets with shifts

We prove that, for any finite set $A \subset \mathbb Q$ with $|AA| \leq K|A|$ and any positive integer $k$, the $k$-fold product set of the shift $A+1$ satisfies the bound $$| \{(a_1+1)(a_2+1) \cdots (a_k+1) : a_i \in A \}| \geq \frac{|A|^k}{(8k^4)^{kK}}. $$ This result is essentially optimal when $K$ is of the order $c\log|A|$, for a sufficiently small constant $c=c(k)$. Our main tool is a multiplicative variant of the $Λ$-constants used in harmonic analysis, applied to Dirichlet polynomials.

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arXiv2017

Bourgain-Chang's proof of the weak Erdős-Szemerédi conjecture

This is an exposition of the following `weak' Erdős-Szemerédi conjecture for integer sets proved by Bourgain and Chang in 2004. For any $γ> 0$ there exists $Λ(γ) > 0$ such that for an arbitrary $A \subset \mathbb{N}$, if $|AA| \leq K|A|$ then $$E_{+}(A) \leq K^Λ|A|^{2+γ}.$$

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arXiv2017

Convex sequences may have thin additive bases

For a fixed $c > 0$ we construct an arbitrarily large set $B$ of size $n$ such that its sum set $B+B$ contains a convex sequence of size $cn^2$, answering a question of Hegarty.

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arXiv2016

On additive bases of sets with small product set

We prove that finite sets of real numbers satisfying $|AA| \leq |A|^{1+ε}$ with sufficiently small $ε> 0$ cannot have small additive bases nor can they be written as a set of sums $B+C$ with $|B|, |C| \geq 2$. The result can be seen as a real analog of the conjecture of Sárközy that multiplicative subgroups of finite fields of prime order are additively irreducible.

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arXiv2016

Discrete spheres and arithmetic progressions in product sets

We prove that if $B$ is a set of $N$ positive integers such that $B\cdot B$ contains an arithmetic progression of length $M$, then for some absolute $C > 0$, $$ π(M) + C \frac {M^{2/3}}{\log^2 M} \leq N, $$ where $π$ is the prime counting function. This improves on previously known bounds of the form $N = Ω(π(M))$ and gives a bound which is sharp up to the second order term, as Pach and Sándor gave an example for which $$ N < π(M)+ O\left(\frac {M^{2/3}}{\log^2 M} \right). $$ The main new tool is a reduction of the original problem to the question of approximate additive decomposition of the $3$-sphere in $\mathbb{F}_3^n$ which is the set of $\{0,1\}$ vectors with exactly three non-zero coordinates. Namely, we prove that such a set cannot have an additive basis of order two of size less than $c n^2$ with absolute constant $c > 0$.

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arXiv2015

On additive shifts of multiplicative almost-subgroups in finite fields

We prove that for sets $A, B, C \subset \mathbb{F}_p$ with $|A|=|B|=|C| \leq \sqrt{p}$ and a fixed $0 \neq d \in \mathbb{F}_p$ holds $$ \max(|AB|, |(A+d)C|) \gg|A|^{1+1/26}. $$ In particular, $$ |A(A+1)| \gg |A|^{1 + 1/26} $$ and $$ \max(|AA|, |(A+1)(A+1)|) \gg |A|^{1 + 1/26}. $$ The first estimate improves the bound by Roche-Newton and Jones. In the general case of a field of order $q = p^m$ we obtain similar estimates with the exponent $1+1/559 + o(1)$ under the condition that $AB$ does not have large intersection with any subfield coset, answering a question of Shparlinski. Finally, we prove the estimate $$ \left| \sum_{x \in \mathbb{F}_q} ψ(x^n) \right| \ll q^{\frac{7 - 2δ_2}{8}}n^{\frac{2+2δ_2}{8}} $$ for Gauss sums over $\mathbb{F}_q$, where $ψ$ is a non-trivial additive character and $δ_2 = 1/56 + o(1)$. The estimate gives an improvement over the classical Weil bound when $q^{1/2} \ll n = o\left( q^{29/57 + o(1)} \right)$.

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