Metric decomposability theorems on sets of integers

A set A & SUB;N$\mathcal {A}\subset \mathbb {N}$ is called additively decomposable (resp., asymptotically additively decomposable) if there exist sets B,C & SUB;N$\mathcal {B},\mathcal {C}\subset \mathbb {N}$ of cardinality at least two each such that A=B+C$\mathcal {A}=\mathcal {B}+\mathcal {C}$ (resp., A & UDelta;(B+C)$\mathcal {A}\Delta (\mathcal {B}+\mathcal {C})$ is finite). If none of these properties hold, the set A${\mathcal {A}}$ is called totally primitive. We define Z$\mathbb {Z}$-decomposability analogously with subsets A,B,C$\mathcal {A,B,C}$ of Z$\mathbb {Z}$. Wirsing showed that almost all subsets of N$\mathbb {N}$ 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 Z$\mathbb {Z}$ are Z$\mathbb {Z}$-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.