Abstract
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
$\phi _1, \phi _2, \phi _3: G \to G$
be commuting endomorphisms whose images have finite indices, we show that
(1)
If
$A \subset G$
has positive upper Banach density and
$\phi _1 + \phi _2 + \phi _3 = 0$
, then
$\phi _1(A) + \phi _2(A) + \phi _3(A)$
contains a Bohr set. This generalizes a theorem of Bergelson and Ruzsa in
$\mathbb {Z}$
and a recent result of the first author.
(2)
For any partition
$G = \bigcup _{i=1}^r A_i$
, there exists an
$i \in \{1, \ldots , r\}$
such that
$\phi _1(A_i) + \phi _2(A_i) - \phi _2(A_i)$
contains a Bohr set. This generalizes a result of the second and third authors from
$\mathbb {Z}$
to countable abelian groups.
(3)
If
$B, C \subset G$
have positive upper Banach density and
$G = \bigcup _{i=1}^r A_i$
is a partition,
$B + C + A_i$
contains a Bohr set for some
$i \in \{1, \ldots , r\}$
. This is a strengthening of a theorem of Bergelson, Furstenberg and Weiss.
All results are quantitative in the sense that the radius and rank of the Bohr set obtained depends only on the indices
$[G:\phi _j(G)]$
, the upper Banach density of A (in (1)), or the number of sets in the given partition (in (2) and (3)).
Publisher
Cambridge University Press (CUP)
Subject
Computational Mathematics,Discrete Mathematics and Combinatorics,Geometry and Topology,Mathematical Physics,Statistics and Probability,Algebra and Number Theory,Theoretical Computer Science,Analysis