2026/07/20 by Ioannis Kousek
Mathematics · #math.CO #math.DS
Motivated by recent results \citecharamaraskousekmountakisradic2025BBingroups on infinite sumsets of the form B+B=\b1+b2:b1,b2∈ B\ in large subsets of abelian groups, and an old problem of Owings \cite[Problem E2494]Owingproblems about the partition regularity of B+B in 2 colours, we show the following theorem. Let (G,+) be a countable abelian group such that the subgroup \g+g\colon g∈ G\ has finite index and the doubling map D: g↦ g+g has finite kernel. Let also Φ=(ΦN)N be any Folner sequence in G and Φ/2=(D-1(ΦN))N. Then, if A⊂ G is such that dΦ(A)+dΦ/2(A)>1, there is an infinite set B⊂ G and some t∈ G for which t+B+B⊂ A. We prove that this result implies the main theorem in \citecharamaraskousekmountakisradic2025BBingroups, and construct an example to show the reverse implication does not hold. Moreover, we show that our main theorem is optimal in a strong sense. Namely, for any countable abelian group (G,+) with the aforementioned assumptions -- which are necessary -- there exists a Folner sequence Φ and a set A⊂ G so that dΦ(A)+dΦ/2(A)=1, but there is no infinite set B⊂ G and t∈ G for which t+B+B⊂ A. Finally, we relate the optimality of our main result in the integer setting to Owings' problem and present some other considerations around this.