A density theorem for prime squares

TL;DR

The paper addresses representing large integers as sums of prime squares drawn from a dense subset of primes. It leverages Green's transference principle to reduce to a pseudorandom setting created via the $W$-trick and a modular framework with $W=8\prod_{2<p<w} p$, enabling a density version of Hua's theorem for $s\ge 8$. By constructing majorants $\nu_b$ and bounded functions $f_b$, and proving pseudorandomness and restriction properties through major/minor arc analysis and Gauss-sum estimates, the authors establish that every sufficiently large $n$ with $n\equiv s\pmod{24}$ is a sum of $s$ prime squares from $P$ whenever $P$ has lower density $\delta_P>\sqrt{1-\min\{s,16\}/32}$. The approach blends harmonic analysis, additive combinatorics, and analytic number theory to extend density results to prime-square representations, offering a robust framework for similar problems with sparse prime structures.

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 $P$.

Theorems & Definitions (24)

• Theorem 1.1
• Lemma 3.1: Transference Lemma
• proof
• Proposition 3.2
• Proposition 3.3
• Lemma 3.4
• proof
• proof : Proof of Theorem 1.1
• Proposition 4.1
• Lemma 4.2