A conjecture of S\'ark\"ozy on quadratic residues, II

Denote by $\mathcal{R}_p$ the set of all quadratic residues in $\mathbf{F}_p$ for each prime $p$. A conjecture of A. S\'ark\"ozy asserts, for all sufficiently large $p$, that no subsets $\mathcal{A},\mathcal{B}\subseteq\mathbf{F}_p$ with $|\mathcal{A}|,|\mathcal{B}|\geqslant2$ 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.