A density theorem for prime squares

Abstract

Let $s\ge 8$ 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 $n\equiv s(\mathrm{mod}24)$ can be represented by a sum of $s$ squares of primes in $P$.

15 pages