D. Zakharov

activity

20162026

most citedEssentially tight bounds for rainbow cycles in proper edge-colourings

3 citations · 15 across the 35 of their papers we have counts for

collaborators

36 papers

math.CA2026

Kakeya and Furstenberg problems for sticky sets of tubes

Hong Wang, Dmitrii Zakharov

We study incidence problems for sticky sets of tubes. In the first part, we prove a sharp Furstenberg set estimate for sticky sets of tubes in $R^{3}$ satisfying convex Wolff…

math.CA2026

Estimating the number of real zeros of linear combinations of radicals of polynomials

Gal Binyamini, Avner Kiro, Alexander Logunov +2

We obtain upper bounds for the number of real zeros of functions of the form $f (x) = k = 1 \sum n c_{k} (P_{k} (x))^{α_{k}},$ where $c_{k}, α_{k} \in R$ and each $P_k…

math.CO2026

The Erdős distinct distances problem in $R^{3}$

Jonathan Tidor, Hung-Hsun Hans Yu, Dmitrii Zakharov

We prove that $N$ points in $R^{3}$ determine at least $N^{2/3 - o (1)}$ distinct distances.

math.CO2026

Asymptotically sharp bounds for affine subspace statistics in $F_{2}^{n}$

Ting-Wei Chao, Zixuan Xu, Dmitrii Zakharov

Given a subset $A \subseteq F_{2}^{n}$ , we can consider the distribution of the intersection size of $A$ with a uniformly random $d$ -flat $F$ . Motivated by the edge statistics…

math.CO2026

Uniform Set Systems with Uniform Witnesses

Ting-Wei Chao, Zixuan Xu, Dmitrii Zakharov

Frankl--Pach and Erdős conjectured that any $(d + 1)$ -uniform set family $F \subseteq (d + 1 [ n ])$ with VC-dimension at most $d$ has size at most $(d n - 1)$ when…

math.CO2026

An isoperimetric inequality for word overlap

Dmitrii Zakharov

Let $A$ and $B$ be sets of words of length $n$ over some finite alphabet. Suppose that no suffix of a word in $A$ coincides with a prefix of a word in $B$ . Then we show that the pr…

math.CO2025

Moderate-doubling sets in $F_{2}^{n}$ intersect subspaces

Alex Cohen, Dmitrii Zakharov

We show that any set $A$ in $F_{2}^{n}$ with $∣ A + A ∣ \leq ∣ A ∣^{2 - η}$ must intersect a subspace of dimension $O_{η} (lo g ∣ A ∣)$ in at least $∣ A ∣^{η - o (1)}$ elements.

math.CO2025

A fractal-like configuration of point-line pairs for the minimal distance problem

Alexander Logunov, Dmitrii Zakharov

We show that for every $n \in N$ there is a collection of points $p_{1}, \dots, p_{n}$ and lines $ℓ_{1}, \dots, ℓ_{n}$ in the unit square such that for any $i$ we have $p_i…

math.CO2025

Heilbronn's triangle problem in three dimensions

Dominique Maldague, Hong Wang, Dmitrii Zakharov

We show that among any $n$ points in the unit cube one can find a triangle of area at most $n^{- 2/3 - c}$ for some absolute constant $c > 0$ . This gives the first non-trivial upper bo…

math.CA2025

A Continuum Beck-type Theorem for Hyperplanes

Paige Bright, Alexander Ortiz, Dmitrii Zakharov

We prove a sharp continuum Beck-type theorem for hyperplanes. Our work is inspired by foundational work of Beck on the discrete problem, as well as refinements due to Do and Lund.…

math.CO2024

Generalized Arithmetic Kakeya

Cosmin Pohoata, Dmitrii Zakharov

Around the early 2000-s, Bourgain, Katz and Tao introduced an arithmetic approach to study Kakeya-type problems. They showed that the Euclidean Kakeya conjecture follows from a nat…

math.CO2024

Color avoidance for monotone paths

Eion Mulrenin, Cosmin Pohoata, Dmitrii Zakharov

In 2014, Moshkovitz and Shapira determined the tower height for hypergraph Ramsey numbers of tight monotone paths. We address the color-avoiding version of this problem in which on…

math.CO2024

Sharp bound for the Erdős-Straus non-averaging set problem

Huy Tuan Pham, Dmitrii Zakharov

A set of integers $A$ is non-averaging if there is no element $a$ in $A$ which can be written as an average of a subset of $A$ not containing $a$ . We show that the largest non-aver…

math.CO2024

Ruzsa's problem on Bi-Sidon sets

János Pach, Dmitrii Zakharov

A subset $S$ of real numbers is called bi-Sidon if it is a Sidon set with respect to both addition and multiplication, i.e., if all pairwise sums and all pairwise products of eleme…

math.CO2024

Lower bounds for incidences

Alex Cohen, Cosmin Pohoata, Dmitrii Zakharov

Let $p_{1}, \dots, p_{n}$ be a set of points in the unit square and let $T_{1}, \dots, T_{n}$ be a set of $δ$ -tubes such that $T_{j}$ passes through $p_{j}$ . We prove a lower bound for the numbe…

math.NT2024

On the minimal period of integer tilings

Izabella Łaba, Dmitrii Zakharov

If a finite set $A$ tiles the integers by translations, it also admits a tiling whose period $M$ has the same prime factors as $∣ A ∣$ . We prove that the minimal period of such a til…

math.CO2024

An explicit economical additive basis

Vishesh Jain, Huy Tuan Pham, Mehtaab Sawhney +1

We present an explicit subset $A \subseteq N = {0, 1, \dots}$ such that $A + A = N$ and for all $ε > 0$ , \[\lim_{N\to \infty}\frac{\big|\big\{(n_1,n_2):…

math.CA2024★ 1 cited

On sets of orthogonal exponentials on the disk

Dmitrii Zakharov

We show that if $A$ is a set of mutually orthogonal exponentials with respect to the unit disk then $∣ A \cap [- R, R]^{2} ∣ ≲_{ε} R^{3/5 + ε}$ holds. This improv…

math.NT2024

Most integers are not a sum of two palindromes

Dmitrii Zakharov

For $g \geq 2$ , we show that the number of positive integers at most $X$ which can be written as sum of two base $g$ palindromes is at most $\frac{X}{l o g ^{c} X}$ . This answers a ques…

math.CO2024

On skew corner-free sets

Cosmin Pohoata, Dmitrii Zakharov

We construct skew corner-free sets in $[n]^{2}$ of size $n^{5/4}$ , thereby disproving a conjecture of Kevin Pratt. We also show that any skew corner-free set in $\mathbb{F}_{q}^{n} \…

math.MG2023

Spherical sets avoiding orthonormal bases

Dmitrii Zakharov

We show that there exists an absolute constant $c_{0} < 1$ such that for all $n \geq 2$ , any measurable set $A \subset S^{n - 1}$ of density at least $c_{0}$ contains $n$ pairwise orthogona…

math.CO2023

A Sharp Ramsey Theorem for Ordered Hypergraph Matchings

Lisa Sauermann, Dmitrii Zakharov

We prove essentially sharp bounds for Ramsey numbers of ordered hypergraph matchings, inroduced recently by Dudek, Grytczuk, and Ruciński. Namely, for any $r \geq 2$ and $n \geq 2$ ,…

math.CO2023★ 3 cited

Essentially tight bounds for rainbow cycles in proper edge-colourings

Noga Alon, Matija Bucić, Lisa Sauermann +2

An edge-coloured graph is said to be rainbow if no colour appears more than once. Extremal problems involving rainbow objects have been a focus of much research over the last decad…

math.CO2023★ 1 cited

A new upper bound for the Heilbronn triangle problem

Alex Cohen, Cosmin Pohoata, Dmitrii Zakharov

For sufficiently large $n$ , we show that in every configuration of $n$ points chosen inside the unit square there exists a triangle of area less than $n^{- 8/7 - 1/2000}$ . This improv…

Kakeya and Furstenberg problems for sticky sets of tubes

Estimating the number of real zeros of linear combinations of radicals of polynomials

The Erdős distinct distances problem in R3

Asymptotically sharp bounds for affine subspace statistics in F2n​

Uniform Set Systems with Uniform Witnesses

An isoperimetric inequality for word overlap

Moderate-doubling sets in F2n​ intersect subspaces

A fractal-like configuration of point-line pairs for the minimal distance problem

Heilbronn's triangle problem in three dimensions

A Continuum Beck-type Theorem for Hyperplanes

Generalized Arithmetic Kakeya

Color avoidance for monotone paths

Sharp bound for the Erdős-Straus non-averaging set problem

Ruzsa's problem on Bi-Sidon sets

Lower bounds for incidences

On the minimal period of integer tilings

An explicit economical additive basis

On sets of orthogonal exponentials on the disk

Most integers are not a sum of two palindromes

On skew corner-free sets

Spherical sets avoiding orthonormal bases

A Sharp Ramsey Theorem for Ordered Hypergraph Matchings

Essentially tight bounds for rainbow cycles in proper edge-colourings

A new upper bound for the Heilbronn triangle problem

The Erdős distinct distances problem in $R^{3}$

Asymptotically sharp bounds for affine subspace statistics in $F_{2}^{n}$

Moderate-doubling sets in $F_{2}^{n}$ intersect subspaces