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arXiv · 2604.12864

An inverse theorem for sumsets of sets of positive density in the integers

Ethan Ackelsberg·Florian K. Richter
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Abstract

Let $d(\cdot)$ denote the natural density on the positive integers. We characterize all sets $A,B$ with positive density satisfying $d(A+B)=d(A)+d(B)$, under the assumption that the two sets are not both contained in a proper finite union of residue classes. This gives a new inverse theorem for Kneser's sumset inequality in the integers, and provides a partial answer to a long-standing open question of Erd\H{o}s and Graham.

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BibTeXRIS

Ethan Ackelsberg, Florian K. Richter. 2026-04-14. An inverse theorem for sumsets of sets of positive density in the integers. https://arxiv.org/abs/2604.12864

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