Subsets of \(\mathbb{F}_p^*\) with only small products or ratios

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