Metric decomposability theorems on sets of integers

A set (Formula presented.) is called additively decomposable (resp., asymptotically additively decomposable) if there exist sets (Formula presented.) of cardinality at least two each such that (Formula presented.) (resp., (Formula presented.) is finite). If none of these properties hold, the set (Formula presented.) is called totally primitive. We define (Formula presented.) -decomposability analogously with subsets (Formula presented.) of (Formula presented.). Wirsing showed that almost all subsets of (Formula presented.) 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 (Formula presented.) are (Formula presented.) -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.