# Refined Estimates Concerning Sumsets Contained in the Roots of Unity

## TL;DR

This paper establishes new bounds on the clique number of Paley graphs and characterizes additive decompositions of quadratic residues, advancing understanding of their combinatorial structure.

## Contribution

It provides refined upper bounds for the clique number of Paley graphs and characterizes additive decompositions of quadratic residues as co-Sidon sets.

## Key findings

- Clique number of Paley graph at most sqrt{p/2} + 1
- Additive decompositions of quadratic residues are limited to co-Sidon sets
- Enhanced understanding of sumset structures in roots of unity

## Abstract

We prove that the clique number of the Paley graph is at most $\sqrt{p/2} + 1$, and that any supposed additive decompositions of the set of quadratic residues can only come from co-Sidon sets.