We found a match
Your institution may have rights to this item. Sign in to continue.
- Title
Bohr sets in sumsets II: countable abelian groups.
- Authors
Griesmer, John T.; Le, Anh N.; Thái Hoàng Lê
- Abstract
We prove three results concerning the existence of Bohr sets in threefold sumsets. More precisely, letting G be a countable discrete abelian group and ϕ1,ϕ2,ϕ3:G→G be commuting endomorphisms whose images have finite indices, we show that (1) If A⊂G has positive upper Banach density and ϕ1+ϕ2+ϕ3=0, then ϕ1(A)+ϕ2(A)+ϕ3(A) contains a Bohr set. This generalizes a theorem of Bergelson and Ruzsa in Z and a recent result of the first author. (2) For any partition G=⋃ri=1Ai, there exists an i∈{1,...,r} such that ϕ1(Ai)+ϕ2(Ai)-ϕ2(Ai) contains a Bohr set. This generalizes a result of the second and third authors from Z to countable abelian groups. (3) If B,C⊂G have positive upper Banach density and G=⋃ri=1Ai is a partition, B+C+Ai contains a Bohr set for some i∈{1,...,r}. This is a strengthening of a theorem of Bergelson, Furstenberg, and Weiss. These results are quantitative in the sense that the radius and rank of the Bohr set obtained depends only on the indices [G:ϕj(G)], the upper Banach density of A (in (1)), or the number of sets in the given partition (in (2) and (3)).
- Subjects
ABELIAN groups; ENDOMORPHISMS; ARTIFICIAL intelligence
- Publication
Forum of Mathematics, Sigma, 2023, Vol 11, p1
- ISSN
2050-5094
- Publication type
Article
- DOI
10.1017/fms.2023.49