On the density version of quadratic Waring's problem and the quadratic Waring--Goldbach problem

TL;DR

This paper establishes a sharp density theorem for quadratic Waring's problem over cyclic groups with at least five variables and improves results on the density version of the quadratic Waring-Goldbach problem over integers.

Contribution

It introduces a precise density theorem for quadratic Waring's problem over cyclic groups and advances the understanding of the quadratic Waring-Goldbach problem over integers.

Findings

Proves a sharp density theorem for quadratic Waring's problem over cyclic groups.

Provides new improvements on the density version of the quadratic Waring-Goldbach problem.

Achieves results for problems with at least five variables.

Abstract

We prove a sharp density theorem for quadratic Waring's problem over cyclic groups, when the number of variables is at least $5$. Also, we obtain some new improvements on the density version of the quadratic Waring--Goldbach problem over integers.