Counting additive decompositions of quadratic residues in finite fields

Abstract

We say that a set $\mathcal{S}$ is additively decomposed into two sets $\mathcal{A}$ and $\mathcal{B}$ if $\mathcal{S} = \{a+b: a\in \mathcal{A}, b \in \mathcal{B}\}$. A. Sárközy has recently conjectured that the set $\mathcal{Q}$ of quadratic residues modulo a prime $p$ does not have nontrivial decompositions. Although various partial results towards this conjecture have been obtained, it is still open. Here we obtain a nontrivial upper bound on the number of such decompositions.