Metric decomposability theorems on sets of integers

2022/04/25 by Bienvenu, P. -Y.
#FOS: Mathematics #Number Theory (math.NT) #Probability (math.PR)

paper · doi:10.48550/arxiv.2204.11773

Abstract

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

Related

• Metric decomposability theorems on sets of integers
• On additive and multiplicative decompositions of sets of integers with restricted prime factors, I. (Smooth numbers.)
• On additive and multiplicative decompositions of sets of integers with restricted prime factors, II. (Smooth numbers and generalizations.)
• The structure of decomposable lattices determined by their prime ideals
• Sets of Completely Decomposed Primes in Extensions of Number Fields