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

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 $|\A|=\alpha N$.