Subsets of ${\mathbb{F}}_p^{\ast }$ with only small products or ratios

Abstract

Let $p$ be a fixed prime. We estimate the number of elements of a set $A\subseteq {\mathbb{F}}_p^{\ast }$ for which $$ s_1s_2 \equiv a \pmod{p} \quad \mbox{for some}\quad a \in [-X,X] \quad \mbox{for all}\quad s_1,s_2 \in A. $$ We also consider variations and generalizations.