A density theorem for prime squares

TL;DR

This paper proves that under certain density conditions on a set of primes, all sufficiently large integers congruent to a specific residue modulo 24 can be expressed as sums of prime squares from that set.

Contribution

It establishes a density theorem for prime squares, extending the understanding of additive representations involving primes with specified density.

Findings

Sufficiently large integers in certain residue classes can be represented as sums of prime squares.

Sets of primes with density above a specific threshold enable such representations.

The result applies to primes with relative lower density exceeding a calculated bound.

Abstract

Let $s\geq 8$ be an integer and $P$ be a set of primes with relative lower density greater than $\sqrt{1-\min\{s,16\}/32}$. We prove that every sufficiently large integer $n\equiv s({\rm mod}24)$ can be represented by a sum of $s$ squares of primes in $P$.