Infinite unrestricted sumsets in subsets of abelian groups with large density

2025/04/11 by Dimitrios Charamaras, Ioannis Kousek, Charamaras, Dimitrios +5 · 2 citations
Mathematics · #05D10 (Primary) 11B13 #11B30 (Secondary) #37A15 #Combinatorics (math.CO) #Dynamical Systems (math.DS) #FOS: Mathematics #Limits and Structures in Graph Theory

paper · pdf · doi:10.48550/arxiv.2504.08649

openalex publication_date 2025/04/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let (G,+) be a countable abelian group such that the subgroup \g+g\colon g∈ G\ has finite index and the doubling map g↦ g+g has finite kernel. We establish lower bounds on the upper density of a set A⊂ G with respect to an appropriate Følner sequence, so that A contains a sumset of the form \t+b₁+b₂\colon b₁,b₂∈ B\ or \b₁+b₂\colon b₁,b₂∈ B\, for some infinite B⊂ G and some t∈ G. Both assumptions on G are necessary for our results to be true. We also characterize the Følner sequences for which this is possible. Finally, we show that our lower bounds are optimal in a strong sense.

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• On density analogs of Hindman's finite sums theorem
• Infinite linear patterns in sets of positive density

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