The large sieve inequality and additive decompositions of sums of squares

Abstract

Building on the large sieve, we prove a new sieve estimate for sets which are missing various residue classes modulo prime squares. For certain interesting residue class configurations, our bound is a significant improvement over the corresponding Johnsen-Selberg sieve inequality. Our application of this sieve estimate is motivated by Ostmann's problem. We study additive decompositions of sums of squares and show that any non-trivial sumset decomposition must consist of two sets of roughly equal size: If $A_1, A_2 \subseteq \mathbb{Z}$ with $A_1 + A_2 = \{x^2 + y^2 : x, y \in \mathbb{Z}\}$ and $|A_1|, |A_2| \geq 2$, then $\sqrt{x}/(\log x)^{9/2} \ll |A_i \cap [1, x]| \ll \sqrt{x}(\log x)^4$ for $i \in \{1, 2\}$.