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

TL;DR

This paper disproves Sárközy's conjecture by constructing a set in a finite field that defies the expected sum and product coverage, establishing the exact threshold at half the field size.

Contribution

It provides a counterexample to Sárközy's conjecture, showing the threshold for sum-product coverage in finite fields is exactly half the field size.

Findings

Counterexample exists for all odd primes p ≥ 5.

Sets of size (p-1)/2 can exclude 1 from A*.

The threshold for coverage is exactly p/2.

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 $\frac12$.