Sumsets in quadratic residues

Abstract

We describe all sets $A \subseteq \F_p$ which represent the quadratic residues $R \subseteq \F_p$ as $R=A+A$ and $R=A\widehat{+}A$. Also, we consider the case of an approximate equality $R\approx A+A$ and $R\approx A\widehat{+}A$ and prove that $A$ has a structure in the situation.

21 pages