Almost all sets of nonnegative integers and their small perturbations are not sumsets

Abstract

Fix α ∈ ( 0 , 1 / 3 ) \alpha \in (0,1/3) . We show that, from a topological point of view, almost all sets A ⊆ N A\subseteq \mathbb {N} have the property that, if A ′ = A A^\prime =A for all but o ( n α ) o(n^{\alpha }) elements, then A ′ A^\prime is not a nontrivial sumset B + C B+C . In particular, almost all A A are totally irreducible. In addition, we prove that the measure analogue holds with α = 1 \alpha =1 .