The large sieve inequality and additive decompositions of sums of squares

Christian Elsholtz; Julia Stadlmann

paper

The large sieve inequality and additive decompositions of sums of squares

arXiv:2610.20554

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 $A_{1}, A_{2} \subseteq Z$ with $A_{1} + A_{2} = {x^{2} + y^{2} : x, y \in Z}$ and $∣ A_{1} ∣, ∣ A_{2} ∣ \geq 2$ , then $x / (lo g x)^{9/2} ≪ ∣ A_{i} \cap [1, x] ∣ ≪ x (lo g x)^{4}$ for $i \in {1, 2}$ .