D. Zakharov

activity

20242026

collaborators

6 papers

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 ErdÅs 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.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 $∣ A + A ∣ \leq ∣ A ∣^{2 - I^\cdot}$ must intersect a subspace of dimension $O_{I^} \cdot (lo g ∣ A ∣)$ in at least $∣ A ∣^{I^\cdot - o (1)}$ 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.CA2024

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…

Asymptotically sharp bounds for affine subspace statistics in F2n​

Uniform Set Systems with Uniform Witnesses

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

On sets of orthogonal exponentials on the disk

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

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