Metric decomposability theorems on sets of integers

Abstract

AbstractA set is called additively decomposable (resp., asymptotically additively decomposable) if there exist sets of cardinality at least two each such that (resp., is finite). If none of these properties hold, the set is called totally primitive. We define ‐decomposability analogously with subsets of . Wirsing showed that almost all subsets of are totally primitive. In this paper, in the spirit of Wirsing, we study decomposability from a probabilistic viewpoint. First, we show that almost all symmetric subsets of are ‐decomposable. Then we show that almost all small perturbations of the set of primes yield a totally primitive set. Further, this last result still holds when the set of primes is replaced by the set of sums of two squares, which is by definition decomposable.