Sumsets in quadratic residues

TL;DR

This paper characterizes all subsets of finite fields that sum to quadratic residues and explores the structure of sets approximately equal to quadratic residues, revealing their underlying structure.

Contribution

It provides a complete classification of sets representing quadratic residues as sums and analyzes the structure of approximately representing sets.

Findings

All sets $A$ with $A+A=R$ or $A ilde{+}A=R$ are described.

Sets approximately equal to quadratic residues have a specific structure.

The paper extends understanding of sumset representations of 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 $R=A\hat{+} A$. Also, we consider the case of an approximate equality $R \approx A+A$ and $R \approx A\hat{+} A$ and prove that $A$ has a structure in the situation.