On a question of S\'ark\"ozy on gaps of product sequences

TL;DR

This paper investigates the distribution of gaps in product sequences derived from sets with positive upper Banach density, establishing bounds on the size of these gaps and their optimality, and exploring related quotient set properties.

Contribution

It provides new bounds on the gaps in product sequences with positive density and proves these bounds are optimal, addressing a question posed by Sárközy.

Findings

Infinitely many gaps are bounded by \\alpha^{-3}

Existence of infinitely many t-gaps bounded by t^2 \\alpha^{-4}

Bounds are proven to be optimal

Abstract

Motivated by a question of S\'ark\"ozy, we study the gaps in the product sequence $\B=\A ... \A=\{b_n=a_ia_j, a_i,a_j\in \A\}$ when $\A$ has upper Banach density $\alpha>0$. We prove that there are infinitely many gaps $b_{n+1}-b_n\ll \alpha^{-3}$ and that for $t\ge2$ there are infinitely many $t$-gaps $b_{n+t}-b_{n}\ll t^2\alpha^{-4}$. Furthermore we prove that these estimates are best possible. We also discuss a related question about the cardinality of the quotient set $\A/\A=\{a_i/a_j, a_i,a_j\in \A\}$ when $\A\subset\{1,..., N\}$ and $|\A|=\alpha N$.