A counterexample to a conjecture of S\'ark\"ozy on sums and products modulo a prime

Abstract

Let $p$ be a prime and, for $A\subseteq \mathbb F_p$, define $A^\ast=(A+A)\cup(AA)$. S\'ark\"ozy conjectured that there exist constants $c>0$ and $p_0$ such that, for every prime $p>p_0$, every set $A\subseteq \mathbb F_p$ with $|A|>\left(\frac12-c\right)p$ satisfies $\mathbb F_p^\times\subseteq A^\ast$. We disprove this conjecture: for every odd prime $p\ge 5$, there exists a set $A\subseteq \mathbb F_p$ with $|A|=\frac{p-1}{2}$ such that $1\notin A^\ast$. Thus no positive constant $c$ can satisfy S\'ark\"ozy's conjecture. Conversely, if $|A|>\frac{p}{2}$, then $A+A=\mathbb F_p$. Therefore the sharp threshold is exactly $\frac12$.