We construct a set of integers S such that every translate of S is a set of recurrence and a set of rigidity for a weak mixing measure preserving system. Here "set of rigidity" means that enumerating S as (s n ) n∈N produces a rigidity sequence. This construction generalizes or strengthens results of Katznelson, Saeki (on equidistribution and the Bohr topology), Forrest (on sets of recurrence and strong recurrence), and Fayad and Kanigowski (on rigidity sequences). The construction also provides a density analogue of Julia Wolf's results on popular differences in finite abelian groups.Question 1.2. If d * (A) > 0, is there necessarily a c > 0 such that P c (A) contains a Bohr neighborhood of some n 0 ∈ Z?

We answer a question of Hegyvári 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, and it does not require all addends in the triple products we consider to have positive (left) upper Banach densities; one of the addends is allowed to only have positive upper asymptotic density with respect to a (possibly very sparse) ergodic sequence. MAIN RESULTSand if S is a finite set of finite-dimensional irreducible unitary Γ -representations and ε > 0, we define the Bohr set U S,ε by U S,ε = γ ∈ Γ : |γ| σ < ε, for all σ ∈ S .We denote by Γ FD the set of (unitary equivalence classes) of finite-dimensional irreducible unitary Γ -representations. When Γ is abelian, Γ FD ∼ = Hom(Γ , S 1 ), where S 1 is the multiplicative group of complex numbers of modulus 1; thuswhere S is a finite subset of Hom(Γ , S 1 ). If Γ lacks non-trivial finite-dimensional unitary representations, for instance, if Γ is a finitely generated infinite simple group, then the only Bohr set is Γ itself.Let B be a subset of Γ . Given a sequence (T n ) of finite sets in Γ , we define the upper asymptotic density of B along (T n ) by1) and if Γ is amenable, we define the upper Banach density of B by d * (B) = sup d (F n ) (B) : (F n ) is a left Følner sequence . (1.2)A sequence (T n ) of finite subsets of Γ is called ergodic (or equidistributed) if for every unitary Γ -representation (H, π) and for all u ∈ H, 1 |T n | γ∈T n π(γ)u → P π u,

Abstract. R. Jin showed that whenever A and B are sets of integers having positive upper Banach density, the sumset A + B := {a + b : a ∈ A, b ∈ B} is piecewise syndetic. This result was strengthened by Bergelson, Furstenberg, and Weiss to conclude that A + B must be piecewise Bohr. We generalize the latter result to cases where A has Banach density 0, giving a new proof of the previous results in the process.

We collect problems on recurrence for measure preserving and topological actions of a countable abelian group, considering combinatorial versions of these problems as well. We solve one of these problems by constructing, in G 2 := ∞ n=1 Z/2Z, a set S such that every translate of S is a set of topological recurrence, while S is not a set of measurable recurrence. This construction answers negatively a variant of the following question asked by several authors: if A ⊂ Z has positive upper Banach density, must A − A contain a Bohr neighborhood of some n ∈ Z?We also solve a variant of a problem posed by the author by constructing, for all ε > 0, sets S, A ⊆ G 2 such that every translate of S is a set of topological recurrence, d * (A) > 1−ε, and the sumset S + A is not piecewise syndetic. Here d * denotes upper Banach density.

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.