| Summary: | 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.
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