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

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 $\mathrm{\mid }A\mathrm{\mid }>(\frac{1}{2}-c)p$ 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 $\mathrm{\mid }A\mathrm{\mid }=\frac{p-1}{2}$ such that $1\mathrm{\notin }{A}^{\ast }$. Thus no positive constant $c$ can satisfy S'ark"ozy’s conjecture. Conversely, if $\mathrm{\mid }A\mathrm{\mid }>\frac{p}{2}$, then $A+A={\mathbb{F}}_p$. Therefore the sharp threshold is exactly $\frac{1}{2}$.