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arXiv · 2504.08649

Infinite unrestricted sumsets in subsets of abelian groups with large density

Dimitrios Charamaras·Ioannis Kousek·Andreas Mountakis·Tristán Radić
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Abstract

Let $(G,+)$ be a countable abelian group such that the subgroup $\{g+g\colon g\in G\}$ has finite index and the doubling map $g\mapsto g+g$ has finite kernel. We establish lower bounds on the upper density of a set $A\subset G$ with respect to an appropriate F{\o}lner sequence, so that $A$ contains a sumset of the form $\{t+b_1+b_2\colon b_1,b_2\in B\}$ or $\{b_1+b_2\colon b_1,b_2\in B\}$, for some infinite $B\subset G$ and some $t\in G$. Both assumptions on $G$ are necessary for our results to be true. We also characterize the F{\o}lner sequences for which this is possible. Finally, we show that our lower bounds are optimal in a strong sense.

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Dimitrios Charamaras, Ioannis Kousek, Andreas Mountakis, Tristán Radić. 2025-04-11. Infinite unrestricted sumsets in subsets of abelian groups with large density. https://arxiv.org/abs/2504.08649

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