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

Leonetti, Paolo

doi:10.1090/proc/16392

Almost all sets of nonnegative integers and their small perturbations are not sumsets
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by Paolo Leonetti;

Proc. Amer. Math. Soc. 151 (2023), 3681-3689

DOI: https://doi.org/10.1090/proc/16392

Published electronically: June 6, 2023

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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 $\alpha =1$.

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