  Let $p$ be a fixed prime. We estimate the number of elements of a set $A
\subseteq \mathbb{F}^*_p$ 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.
