Sárközy’s conjecture on quadratic residues

Abstract: For an odd prime number $p$, let $\mathcal R_p\subset \mathbb F_p$ be the set of all non-zero quadratic residues. A. Sárközy conjectured that the set $\mathcal R_p$ does not admit a non-trivial additive decomposition for large enough $p$, i.e. for $p>p_0$ the identity $A+B=\mathcal R_p$ implies $\min(|A|,|B|)=1$. In this talk we present a complete resolution of Sárközy's conjecture. Further, we show that, for a subgroup $G\subset \mathbb F_p^*$, the equality $G\cup\{0\}=A-A$ for some $A$ implies $|G|=2$ or $6$ and if $G=A+B$ non-trivially, then $|A|=|B|=\sqrt{|G|}$.

number theory

Audience: researchers in the topic

Combinatorial and additive number theory seminar (CANT 2026)