Related papers: Bohr neighborhoods in three-fold difference sets

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…

We prove that there is a set of integers $A$ having positive upper Banach density whose difference set $A-A:=\{a-b:a,b\in A\}$ does not contain a Bohr neighborhood of any integer, answering a question asked by Bergelson, Hegyv\'ari, Ruzsa,…

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…

Let $G$ be a discrete abelian group. F{\o}lner showed that if $A \subseteq G$ has positive upper Banach density, then $A - A$ contains an almost Bohr set -- a set of the form $B \setminus E$ where $B$ is a Bohr set and $E$ has zero Banach…

We answer a question of Hegyv\'ari and Ruzsa concerning effective estimates of the Bohr-regularity of certain triple sums of sets with positive upper Banach densities in the integers. Our proof also works for any discrete amenable group,…

For a fixed prime $p$, $\mathbb F_{p}$ denotes the field with $p$ elements, and $\mathbb F_{p}^{\omega}$ denotes the countable direct sum $\bigoplus_{n=1}^{\infty} \mathbb F_{p}$. Viewing $\mathbb F_{p}^{\omega}$ as a countable abelian…

We analyze sumsets A+B = {a+b : a in A, b in B} where A,B are sets of integers, A is infinite, and B has positive upper Banach density. For each k, we show that A+B contains at least the expected density of k-term arithmetic progressions…

R. Jin showed that whenever A and B are sets of integers having positive upper Banach density, the sumset A+B is piecewise syndetic. This result was strengthened by Bergelson, Furstenberg, and Weiss to conclude that A+B must be piecewise…

We study relations between subsets of integers that are large, where large can be interpreted in terms of size (such as a set of positive upper density or a set with bounded gaps) or in terms of additive structure (such as a Bohr set). Bohr…

If $A$ and $B$ are subsets of an abelian group, their sumset is $A+B:=\{a+b:a\in A, b\in B\}$. We study sumsets in discrete abelian groups, where at least one summand has positive upper Banach density. Renling Jin proved that if $A$ and $B$…

Every set $A$ of positive integers with upper Banach density 1 contains an infinite sequence of pairwise disjoint subsets $(B_i)_{i=1}^{\infty}$ such that $B_i$ has upper Banach density 1 for all $i \in \mathbf{N}$ and $\sum_{i\in I} B_i…

We prove that sets with positive upper Banach density in sufficiently large dimensions contain congruent copies of all sufficiently large dilates of three specific higher-dimensional patterns. These patterns are: $2^n$ vertices of a fixed…

Erd\H{o}s conjectured that for any set $A\subseteq \mathbb{N}$ with positive lower asymptotic density, there are infinite sets $B,C\subseteq \mathbb{N}$ such that $B+C\subseteq A$. We verify Erd\H{o}s' conjecture in the case that $A$ has…

We show that any subset of the natural numbers with positive logarithmic Banach density contains a set that is within a factor of two of a geometric progression, improving the bound on a previous result of the authors. Density conditions on…

We give a short proof of the fact that every set of natural numbers with positive upper Banach density contains the sum of two infinite sets. The approach simplifies earlier existing proofs.

We generalize a result by Cook, Magyar, and Pramanik [3] on three-term arithmetic progressions in subsets of $\mathbb{R}^d$ to corners in subsets of $\mathbb{R}^d\times\mathbb{R}^d$. More precisely, if $1<p<\infty$, $p\neq 2$, and $d$ is…

In this paper we show that every set $A \subset \mathbb{N}$ with positive density contains $B+C$ for some pair $B,C$ of infinite subsets of $\mathbb{N}$, settling a conjecture of Erd\H{o}s. The proof features two different decompositions of…

It was shown by V. Bergelson that any set B with positive upper multiplicative density contains nicely intertwined arithmetic and geometric progressions: For each positive integer k there exist integers a,b,d such that $ {b(a+id)^j:i,j…

A subset $R$ of integers is a set of Bohr recurrence if every rotation on $\mathbb{T}^d$ returns arbitrarily close to zero under some non-zero multiple of $R$. We show that the set $\{k!\, 2^m3^n\colon k,m,n\in \mathbb{N}\}$ is a set of…

A theorem of Bogolyubov states that for every dense set $A$ in $\mathbb{Z}_N$ we may find a large Bohr set inside $A+A-A-A$. In this note, motivated by the work on a quantitative inverse theorem for the Gowers $U^4$ norm, we prove a…