A conjecture of Sárközy on quadratic residues, II

Abstract

Denote by ${\mathcal{R}}_p$ the set of all quadratic residues in ${\mathbf{F}}_p$ for each prime $p$. A conjecture of A. Sárközy asserts, for all sufficiently large $p$, that no subsets $\mathcal{A},\mathcal{B}\subseteq {\mathbf{F}}_p$ with $|\mathcal{A}|,|\mathcal{B}|⩾2$ satisfy $\mathcal{A}+\mathcal{B}={\mathcal{R}}_p$. In this paper, we show that if such subsets $\mathcal{A},\mathcal{B}$ do exist, then there are at least $(\log 2{)}^{-1}\sqrt{p}-1.6$ elements in $\mathcal{A}+\mathcal{B}$ that have unique representations and one should have \begin{align*} \frac{1}{4}\sqrt{p}< |\mathcal{A}|,|\mathcal{B}|< 2\sqrt{p}-1. \end{align*} This refines previous bounds obtained by I.E. Shparlinski, I.D. Shkredov, and Y.-G. Chen and X.-H. Yan. Moreover, we also establish bounds for $|\mathcal{A}|,|\mathcal{B}|$ and the additive energy $E(\mathcal{A},\mathcal{B})$ if few elements in $\mathcal{A}+\mathcal{B}$ have unique representations.