arXiv subjects
Publications and source records attributed to 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 $\mathbb R^3$ satisfying convex Wolff axioms. In the second part, we prove the Kakeya set conjecture in $\mathbb R^4$ for sticky sets of tubes satisfying convex Wolff axioms.
We obtain upper bounds for the number of real zeros of functions of the form $$ f(x) = \sum_{k=1}^{n} c_k \bigl(P_k(x)\bigr)^{α_k}, $$ where $c_k, α_k \in \mathbb{R}$ and each $P_k$ is a real polynomial of degree at most $d$ that is non-negative on an interval $I\subset \mathbb{R}$. We improve previously known exponential upper bounds for the number of roots on $I$ to bounds that are polynomial in $n$, linear in $d$, and independent of the exponents $α_k$. For linear combinations of square roots of positive quadratic polynomials on $\mathbb{R}$ we prove the linear bound $2n$, answering a question of N.~Alon. A modification of the argument yields a linear bound for a question of A.~Gabrielov, D.~Novikov, and B.~Shapiro related to Maxwell's conjecture. The article describes two independent approaches: an elementary ODE method in the general case, which also gives a polynomial bound for the number of critical points of one dimensional Gaussian mixtures, and a PDE method for the case of positive quadratic polynomials, which connects the problem to the number of nodal domains of solutions to $Δu + λu = 0$ on the punctured hyperbolic plane. As a byproduct of the second approach, we describe a curious relation between axially symmetric harmonic functions on $\mathbb{R}^3\setminus\{(x,0,0)\}$ and Laplace-Beltrami eigenfunctions on the hyperbolic plane with eigenvalue $1/4$.
We prove that $N$ points in $\mathbb{R}^3$ determine at least $N^{2/3-o(1)}$ distinct distances.
Given a subset $A \subseteq \mathbb F_2^n$, we can consider the distribution of the intersection size of $A$ with a uniformly random $d$-flat $F$. Motivated by the edge statistics problem and the hypercube statistics problem, the affine subspace statistics problem concerns the maximum of $\mathbb{P}[|F\cap A|=s]$ among $A \subseteq \mathbb F_2^n$ for any fixed $s\in\{1,\dots,2^d\}$ over a uniformly random $d$-flat $F$. We use $λ^*(d,s)$ to denote the limit of the maximum when $n$ goes to infinity. In this note, we prove tight bounds for $λ^*(d,s)$ in two different regimes. For $s=j2^k$ where $j$ is a positive odd integer, the best known lower bound construction achieving $λ^*(d,s)\ge 1-2^{-k}$ is due to taking $A$ as the union of $j$ parallel $(n-d+k)$-flats in $\mathbb F_2^n$. Our main result is a matching upper bound with an additive error term of $O(2^{-3k/2})$. We also study the case $s=1$, where we determine $λ^*(d,1)$ exactly. We show that the random construction where each point is included with probability $2^{-d}$ is optimal.
Denote by ${\mathfrak s}({\mathbb F}_p^d)$ the Erd{\H o}s--Ginzburg--Ziv constant of $\mathbb F_p^d$, that is, the minimum $s$ such that every sequence of $s$ vectors in ${\mathbb F}_p^d$ contains $p$ vectors whose sum is zero. Let ${\mathfrak w}({\mathbb F}_p^d)$ be the maximum size of a sequence of vectors $v_1, \ldots, v_s \in {\mathbb F}_p^d$ such that, for all integers $α_1, \ldots, α_s \ge 0$ with sum $p$, we have $α_1 v_1 + \ldots + α_s v_s \neq 0$ unless $α_i = p$ for some $i$. In 1995, Alon--Dubiner proved that $\mathfrak s(\mathbb F_p^d)$ grows linearly in $p$ when $d$ is fixed. In this work, we determine the constant of linearity: for fixed $d$ and growing $p$, we show that ${\mathfrak s}({\mathbb F}_p^d) = (1+o(1)) {\mathfrak w}({\mathbb F}_p^d) p$. Furthermore, for every prime $p$ and every $d$, we show that ${\mathfrak w}({\mathbb F}_p^d) \le {2d-1 \choose d}+1$. In particular, ${\mathfrak s}({\mathbb F}_p^d) \le 4^d p$ for all sufficiently large $p$ and fixed $d$.
Frankl--Pach and Erdős conjectured that any $(d+1)$-uniform set family $\mathcal{F}\subseteq \binom{[n]}{d+1}$ with VC-dimension at most $d$ has size at most $\binom{n-1}{d}$ when $n$ is sufficiently large. Ahlswede and Khachatrian showed that the conjecture is false by giving a counterexample of size $\binom{n-1}{d}+\binom{n-4}{d-2}$. For a set family $\mathcal{F}\subseteq \binom{[n]}{d+1}$, the condition that its VC-dimension is at most $d$ can be reformulated as follows: for any $F\in\mathcal{F}$, there exists a set $B_F\subseteq F$ such that $F\cap F'\neq B_F$ for all $F'\in\mathcal{F}$. In this direction, the first author, Xu, Yip, and Zhang conjectured that the bound $\binom{n-1}{d}$ holds if we further assume that $|B_F|=s$ for every $F\in \mathcal{F}$ and for some fixed $0\leq s\leq d$. The case $s=0$ is exactly the Erdős--Ko--Rado theorem, and the cases $s\in \{1,d\}$ were proved in the paper by the first author, Xu, Yip, and Zhang. In this short note, we show that the conjecture holds when $s\leq d/2$, and the maximal constructions are stars. Moreover, we construct non-star set families of size $\binom{n-1}{d}$ satisfying the condition for $d/2<s\leq d-1$, which suggests that the problem is substantially different in these cases.
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 product of densities of $A$ and $B$ is upper bounded by $(1+o(1))/(en)$. This bound is asymptotically sharp.
We say that a subset $M$ of $\mathbb R^n$ is exponentially Ramsey if there are $ε>0$ and $n_0$ such that $χ(\mathbb R^n,M)\ge(1+ε)^n$ for any $n>n_0$, where $χ(\mathbb R^n,M)$ stands for the minimum number of colors in a coloring of $\mathbb R^n$ such that no copy of $M$ is monochromatic. One important result in Euclidean Ramsey theory is due to Frankl and Rödl, and states the following (under some mild extra conditions): if both $N_1$ and $N_2$ are exponentially Ramsey then so is $N_1\times N_2$. Applied several times to two-point sets, this result implies that any subset of a `hyperrectangle' is exponentially Ramsey. However, generally, such `embeddings' result in very inefficient bounds on the aforementioned $ε$. In this paper, we present another way of combining exponentially Ramsey sets, which gives much better estimates in some important cases. In particular, we show that the chromatic number of $\mathbb R^n$ with a forbidden equilateral triangle satisfies $χ(\mathbb R^n,\triangle)\ge\big(1.0742...+o(1)\big)^n$, greatly improving upon the previous constant $1.0144$. We also obtain similar strong results for regular simplices of larger dimensions, as well as for related geometric Ramsey-type questions in Manhattan norm. We then show that the same technique implies several interesting corollaries in other combinatorial problems. In particular, we give an explicit upper bound on the size of a family $\mathcal F\subset2^{[n]}$ that contains no weak $k$-sunflowers, i.e. no collection of $k$ sets with pairwise intersections of the same size. This bound improves upon previously known results for all $k\ge4$. Finally, we also present a simple deduction of the (other) celebrated Frankl--Rödl theorem from an earlier result of Frankl and Wilson. It gives probably the shortest known proof of Frankl and Rödl result with the most efficient bounds.
We show that any set $A$ in $\mathbb F_2^n$ with $|A+A| \le |A|^{2-η}$ must intersect a subspace of dimension $O_η(\log |A|)$ in at least $|A|^{η- o(1)}$ elements.
We show that for every $n \in \mathbb N$ there is a collection of points $p_1, \ldots, p_n$ and lines $\ell_1, \ldots, \ell_n$ in the unit square such that for any $i$ we have $p_i \in \ell_i$ and the distance from $p_i$ to any other line $\ell_j$ is at least $c n^{γ-1}$ for some universal constants $c, γ>0$. This is better than a trivial construction by a polynomial factor.
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 bound for the three-dimensional version of Heilbronn's triangle problem. This estimate is a consequence of the following result about configurations of point-line pairs in $\mathbb R^3$: for $n \ge 2$ let $p_1, \ldots,p_n \in [0,1]^3$ be a collection of points and let $\ell_i$ be a line through $p_i$ for every $i$ such that $d(p_i, \ell_j) \ge δ$ for all $i\neq j$. Then we have $n \lesssim δ^{-3+γ}$ for some absolute constant $γ>0$. The analogous result about point-line configurations in the plane was previously established by Cohen, Pohoata and the last author.
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. The inductive proof uses recent breakthrough results in projection theory by Orponen--Shmerkin--Wang and Ren, who proved continuum Beck-type theorems for lines in $\mathbb{R}^2$ and $\mathbb{R}^n$.
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 one no longer necessarily seeks a monochromatic subgraph, but rather one which avoids some colors. This problem was previously studied in uniformity two by Loh and by Gowers and Long. We show, in general, that the tower height for such Ramsey numbers requires one less exponential than in the usual setting. The transition occurs at uniformity three, where the usual Ramsey numbers of monotone paths of length $n$ are exponential in $n$, but the color-avoiding Ramsey numbers turn out to be polynomial.
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-averaging subset of $\{1, \ldots, n\}$ has size $n^{1/4+o(1)}$, thus solving the Erdős-Straus problem. We also determine the largest size of a non-averaging set in a $d$-dimensional box for any fixed $d$. Our main tool includes the structure theorem for the set of subset sums due to Conlon, Fox and the first author, together with a result about the structure of a point set in nearly convex position.
We prove essentially sharp bounds for Ramsey numbers of ordered hypergraph matchings, inroduced recently by Dudek, Grytczuk, and Ruciński. Namely, for any $r \ge 2$ and $n \ge 2$, we show that any collection $\mathcal H$ of $n$ pairwise disjoint subsets in $\mathbb Z$ of size $r$ contains a subcollection of size $\lfloor n^{1/(2^r-1)}/2\rfloor$ in which every pair of sets are in the same relative position with respect to the linear ordering on $\mathbb Z$. This improves previous bounds of Dudek-Grytczuk-Ruciński and of Anastos-Jin-Kwan-Sudakov and is sharp up to a factor of $2$. For large $r$, we even obtain such a subcollection of size $\lfloor (1-o(1))\cdot n^{1/(2^r-1)}\rfloor$, which is asymptotically tight (here, the $o(1)$-term tends to zero as $r \to \infty$, regardless of the value of $n$). Furthermore, we prove a multiparameter extension of this result where one wants to find a clique of prescribed size $m_P$ for each relative position pattern $P$. Our bound is sharp for all choices of parameters $m_P$, up to a constant factor depending on $r$ only. This answers questions of Anastos-Jin-Kwan-Sudakov and of Dudek-Grytczuk-Ruciński.
We show that there exists an absolute constant $c_0<1$ such that for all $n \ge 2$, any measurable set $A \subset S^{n-1}$ of density at least $c_0$ contains $n$ pairwise orthogonal vectors. The result is sharp up to the value of the constant $c_0$. Moreover, we show that for all $2\le k \le n$ a set $A$ avoiding $k$ pairwise orthogonal vectors has measure at most $\exp(-c_1 \min\{\sqrt{n}, n/k\})$ for some $c_1>0$. Proofs rely on the harmonic analysis on the sphere and the hypercontractive inequality.
Let $p_1,\ldots,p_n$ be a set of points in the unit square and let $T_1,\ldots,T_n$ be a set of $δ$-tubes such that $T_j$ passes through $p_j$. We prove a lower bound for the number of incidences between the points and tubes under a natural regularity condition (similar to Frostman regularity). As a consequence, we show that in any configuration of points $p_1,\ldots, p_n \in [0,1]^2$ along with a line $\ell_j$ through each point $p_j$, there exist $j\neq k$ for which $d(p_j, \ell_k) \lesssim n^{-2/3+o(1)}$. It follows from the latter result that any set of $n$ points in the unit square contains three points forming a triangle of area at most $n^{-7/6+o(1)}$. This new upper bound for Heilbronn's triangle problem attains the high-low limit established in our previous work arXiv:2305.18253.
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 decade as they capture the essence of a number of interesting problems in a variety of areas. A particularly intensively studied question due to Keevash, Mubayi, Sudakov and Verstraëte from 2007 asks for the maximum possible average degree of a properly edge-coloured graph on $n$ vertices without a rainbow cycle. Improving upon a series of earlier bounds, Tomon proved an upper bound of $(\log n)^{2+o(1)}$ for this question. Very recently, Janzer-Sudakov and Kim-Lee-Liu-Tran independently removed the $o(1)$ term in Tomon's bound, showing a bound of $O(\log^2 n)$. We prove an upper bound of $(\log n)^{1+o(1)}$ for this maximum possible average degree when there is no rainbow cycle. Our result is tight up to the $o(1)$ term, and so it essentially resolves this question. In addition, we observe a connection between this problem and several questions in additive number theory, allowing us to extend existing results on these questions for abelian groups to the case of non-abelian groups.