Density Bounds in Hua's Theorem for Prime Squares

Abstract

We prove a density version of Hua's theorem in the five-variable case: if $\mathcal{P}$ is a subset of the primes with relative lower density greater than $0.82$, then every sufficiently large $n\equiv 5(\mathrm{mod}24)$ is a sum of five squares of primes from $\mathcal{P}$.

arXiv comment57 pages. A formalization, up to the Siegel-Walfisz and Hua's Theorems, is here: https://github.com/lacey-gatech/Density-Bounds-in-Hua-s-Theorem-for-Prime-Squares