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

Motivated by a question of S'ark"ozy, we study the gaps in the product sequence $\text{B}=\text{A}\mathrm{.}\mathrm{.}\mathrm{.}\text{A}=\{b_n=a_i{a}_j,a_i,a_j\in \text{A}\}$ when $\text{A}$ 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 $t\ge 2$ 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 $\text{A}\mathrm{/}\text{A}=\{a_i\mathrm{/}{a}_j,a_i,a_j\in \text{A}\}$ when $\text{A}\subset \{1,\mathrm{.}\mathrm{.}\mathrm{.},N\}$ and $\mathrm{\mid }\text{A}\mathrm{\mid }=\alpha N$.