Metric decomposability theorems on sets of integers

TL;DR

This paper investigates the decomposability properties of sets of integers and their subsets, demonstrating that most symmetric sets are decomposable while small perturbations of primes tend to be primitive, using probabilistic methods.

Contribution

It provides new probabilistic results on the decomposability and primitiveness of integer sets, extending previous work by Wirsing to symmetric sets and prime-related sets.

Findings

Almost all symmetric subsets of integers are -decomposable.

Small perturbations of prime sets are mostly totally primitive.

The result extends to sets of sums of two squares.

Abstract

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