Joint PU/IAS Number Theory

Ostmann’s problem asks if there are sets $A1$ and $A2$ with $|A1|, |A2| > 1$ so that the sumset $A1 + A2$ differs from the set of primes by only finitely many elements. It is believed that no such $A1$ and $A2$ exist, but to date the problem remains open. A major obstacle to the resolution of Ostmann's problem is the treatment of $A1$ and $A2$ which both occupy approximately half the residue classes mod p for large primes p, and an example of such a set are the squares. Motivated by this obstacle, we study additive decompositions of sums of squares.

Although the set of sums of two squares can be written as a sumset in uncountably many different ways, any non-trivial sumset decomposition must consist of two sets of roughly equal size: We show that if $|A1|, |A2| > 1$ and $A1+A2$ is the set of squares, then $sqrt(x)/(log x)^(7/2) << |A1 ∩ [1,x]|, |A2 ∩ [1,x]| << sqrt(x)*(log x)^3$. The key ingredient of our proof is a new large sieve bound for sets which are missing various residue classes modulo prime squares. That bound is a significant improvement over the corresponding Johnsen-Selberg sieve inequality for certain interesting residue class configurations modulo prime squares. This is joint work with Christian Elsholtz.