arXiv subjects
Publications and source records attributed to Dávid Matolcsi.
In this paper we prove that if $A$ and $B$ are infinite subsets of positive integers such that every positive integer $n$ can be written as $n=ab$, $a\in A$, $b\in B$, then $\displaystyle \lim_{x\to \infty}\frac{A(x)B(x)}{x}=\infty $. We also prove many other results about sets like this.
We discuss the relation of tiling, weak tiling and spectral sets in finite abelian groups. In particular, in elementary $p$-groups $(\mathbb{Z}_p)^d$, we introduce an averaging procedure that leads to a natural object of study: a 4-tuple of functions which can be regarded as a common generalization of tiles and spectral sets. We characterize such 4-tuples for $d=1, 2$, and prove some partial results for $d=3$.
In this paper, we study how dense a multiplicative basis of order $h$ for $\mathbb{Z}^+\!$ can be, improving on earlier results. Upon introducing the notion of a \textit{multiplicative complement}, we present some tight density bounds.
We prove an asymptotic result on the maximum number of k-vertex subtrees in binary trees of given order. This problem turns out to be equivalent to determine the maximum number of k+2-cycles in n-vertex outerplanar graphs, thus we settle the generalized outerplanar Turán number for all cycles. We also determine the exponential growth of the generalized outerplanar Turán number of paths Pk as a function of k which implies the order of magnitude of the generalized outerplanar Turán number of arbitrary trees. The bounds are strongly related to the sequence of Catalan numbers.
In this paper we show that the largest possible size of a subset of $\mathbb{F}_q^n$ avoiding right angles, that is, distinct vectors $x,y,z$ such that $x-z$ and $y-z$ are perpendicular to each other is at most $O(n^{q-2})$. This improves on the previously best known bound due to Naslund \cite{Naslund} and refutes a conjecture of Ge and Shangguan \cite{Ge}. A lower bound of $n^{q/3}$ is also presented. It is also shown that a subset of $\mathbb{F}_q^n$ avoiding triangles with all right angles can have size at most $O(n^{2q-2})$. Furthermore, asymptotically tight bounds are given for the largest possible size of a subset $A\subseteq \mathbb{F}_q^n$ for which $x-y$ is not self-orthogonal for any distinct $x,y\in A$. The exact answer is determined for $q=3$ and $n\equiv 2\pmod {3}$. Our methods can also be used to bound the maximum possible size of a binary code where no two codewords have Hamming distance divisible by a fixed prime $q$. Our lower- and upper bounds are asymptotically tight and both are sharp in infinitely many cases.
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.
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.