Pierre-Yves Bienvenu

Metric decomposability theorems on sets of integers

A set A of integers called additively decomposable (resp. asymptotically additively decomposable) if there exist sets B,C of integers of cardinality at least two each such that A=B+C (resp. the symmetric difference of A and B+C is finite). If none of these properties hold, the set A is called totally primitive. Wirsing showed that almost all sets of integers are totally primitive. In this talk, in the spirit of Wirsing, we study decomposability from a probabilistic viewpoint. We show that almost all small perturbations of the set of primes yield a totally primitive set; ideally this should be true without any perturbation at all (Ostmann's conjecture). Further, this result still holds when the set of primes is replaced by the set of sums of two squares, which is by definition decomposable.