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.