Counting Additive Decompositions of Quadratic Residues in Finite Fields

Abstract

We say that a set $S$ is additively decomposed into two sets $A$ and $B$ if $S=\{a+b:a\in A,b\in B\}$. A. Sárközy has recently conjectured that the set $Q$ 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.