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

TL;DR

This paper demonstrates that most sets of nonnegative integers are structurally resistant to being sumsets, especially under small perturbations, highlighting their inherent irreducibility in a topological and measure-theoretic sense.

Contribution

It establishes that almost all sets are totally irreducible and resistant to becoming sumsets after small modifications, with results in both topological and measure contexts.

Findings

Almost all sets are totally irreducible.

Small perturbations do not produce sumsets.

Results hold in both topological and measure senses.

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 $\alpha=1$.