Bohr Neighborhoods in Generalized Difference Sets

Abstract

If $A$ is a set of integers having positive upper Banach density and $r,s,t$ are nonzero integers whose sum is zero, a theorem of Bergelson and Ruzsa says that the set $rA+sA+tA:=\{ra_1+sa_2+ta_3:a_i\in A\}$ contains a Bohr neighborhood of zero. We prove a natural generalization of this result for subsets of countable abelian groups and more summands.

Bohr sets in sumsets II: countable abelian groups

“…In this paper we extend many of the preceding results to the setting of countable discrete abelian groups. Our main results are discrete analogues of Theorem D, and as such are direct generalizations of Theorems B and C. • In the special case φ j (x) = s j x where s j ∈ Z \ {0}, Theorem 1.2 was proven by the first author [22] without the conclusion on the uniformity of k and η. • The conclusion of Theorem 1.2 remains valid if the φ j do not necessarily commute, but one of them is an automorphism.…”

“…In the spirit of Ruzsa-Hegyvári's result [27] on Bohr sets in A + A − A − a mentioned in the introduction, it is interesting to know if the Bohr set in Theorem 1.7 can be given by a fixed element of C. More precisely, we ask: Our Theorem 1.2 generalizes Theorem B in two ways: replacing the ambient group Z with an arbitrary countable abelian group, and replacing the endomorphisms g → s i g with commuting endomorphisms having finite index image. The main result of [22] generalizes Theorem B in a different way: the endomorphisms still have the form g → s i g, but more summands are considered. The following conjecture is a natural joint generalization of these results.…”

“…In view of Lemma 3.2, let φi : Z → Z be continuous endomorphisms satisfying τ •φ i = φi •τ . Applying this identity to (22), we have…”

“…However, it is not true in general that ψ(G) has finite index (for example, take d = 4, φ 3 = −φ 4 ), and so Conjecture 11.5 is genuinely interesting. It may be necessary to impose some additional hypotheses on the φ j ; see [22,Section 4] for more discussion. Along the same line, we have the following conjecture for partition that extends Theorem 1.4.…”

“…Then φ 1 (A) + φ 2 (A) + φ 3 (A) = A − A, and the Kriz example [31] shows that there exists a set A of positive upper Banach density such that A − A does not contain any Bohr set. See [22,Remark 1.6] for further discussion.…”