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

TL;DR

This paper investigates the structure of subsets in finite fields related to quadratic residues, providing bounds and properties of sumsets under Sárközy's conjecture, refining previous results and exploring additive energies.

Contribution

It offers new bounds on the sizes of subsets whose sumset equals quadratic residues, and analyzes the number of uniquely representable elements, advancing understanding of Sárközy's conjecture.

Findings

If such subsets exist, their sumset has many uniquely representable elements.

Bounds on subset sizes are refined to between quarter and twice the square root of p.

Bounds on additive energy are established for subsets with few unique sum representations.

Abstract

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.