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

Abstract

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

arXiv commentAccepted in Proceedings of the American Mathematical Society