Bohr Sets in Triple Products of Large Sets in Amenable Groups

Abstract

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,

On Katznelson's Question for skew product systems

“…Much more recently, Bergelson and Ruzsa [BR09] showed that the triple sumset r•A+s•A+t•A contains a Bohr neighborhood of zero when r, s, and t are integers with r +s+t = 0 and A is a set of positive upper asymptotic density. Uniformity in the dimension and diameter of Bohr sets contained in triple sumsets was recently demonstrated in broad generality by Björklund and Griesmer [BG19].…”

Bohr neighborhoods in generalized difference sets

“…For example, in Z, the set A of odd integers has d * (A) = 1/2, and A − A + A = A, which does not contain a Bohr neighborhood of 0. Nevertheless, A − A + A contains Bohr neighborhoods of many elements of A whenever d * (A) > 0; see [19], [8].…”

Bohr sets in sumsets II: countable abelian groups

“…Hegyvári and Ruzsa [27] generalized Bogolyubov's theorem in a different direction, showing that there exist "many" a ∈ Z for which A − A + A − a contains a Bohr set. Björklund and the first author [10,Theorem 1.1] strengthened this result by providing explicit bounds on the rank and radius of such a Bohr set, and generalized the result to all countable amenable discrete groups (and hence all countable discrete abelian groups).…”

“…It is well known that all locally compact abelian groups are amenable. Følner [14,15] generalized Theorem A to discrete abelian groups, and the results of [10] mentioned above apply to countable discrete amenable groups which are not necessarily abelian. Against this backdrop, our objective in this program is threefold.…”