Khintchine-type recurrence for 3-point configurations

Abstract

The goal of this paper is to generalize, refine, and improve results on large intersections from [BHK05, ABB21]. We show that if G is a countable abelian group and ϕ, ψ : G → G are homomorphisms such that at least two of the three subgroups ϕ(G), ψ(G), and (ψ − ϕ)(G) have finite index in G, then {ϕ, ψ} has the large intersections property. That is, for any ergodic measure preserving system X = (X, X , µ, (T g ) g∈G ), any A ∈ X , and any ε > 0, the set (Theorem 1.11). Moreover, in the special case where ϕ(g) = ag and ψ(g) = bg for a, b ∈ Z, we show that we only need one of the groups aG, bG, or (b − a)G to be of finite index in G (Theorem 1.13), and we show that the property fails in general if all three groups are of infinite index (Theorem 1.14).One particularly interesting case is where G = (Q >0 , •) and ϕ(g) = g, ψ(g) = g 2 , which leads to a multiplicative version for the Khintchine-type recurrence result in [BHK05]. We also completely characterize the pairs of homomorphisms ϕ, ψ that have the large intersections property when G = Z 2 .The proofs of our main results rely on analysis of the structure of the universal characteristic factor for the multiple ergodic averages 1In the case where G is finitely-generated, the characteristic factor for such averages is the Kronecker factor. In this paper, we study actions of groups that are not necessarily finitely-generated, showing in particular that, by passing to an extension of X, one can describe the characteristic factor in terms of the Conze-Lesigne factor and the σ-algebras of ϕ(G) and ψ(G) invariant functions (Theorem 4.10). ContentsE. ACKELSBERG, V. BERGELSON, AND O. SHALOM 1.6. Structure of the paper 2. Preliminaries 2.1. Uniform Cesàro limits 2.2. Factors 2.3. Factor of invariant sets 2.4. Host-Kra factors 2.5. Joins and meets of factors 2.6. Ziegler factors 2.7. Seminorms for multiplicative configurations 3. Theorem 1.11 3.1. Characteristic factors 3.2. Relative orthonormal basis 3.3. Proof of Theorem 1.11 4. Extensions 4.1. Characteristic factors related to Theorem 1.13 5. A limit formula for {ag, bg} 5.1. Previous limit formulas 5.2. Mackey group 5.3. Cocycle identities 5.4. Proof of Proposition 5.3 5.5. Limit formula 5.6. Proof of Theorem 1.13 6. Proof of Theorem 1.14 7. 3-point configurations in Z 2 7.1. Ergodic popular difference densities when r(M 1 , M 2 ) = (2, 1, 1) 7.2. Ergodic popular difference densities when r(M 1 , M 2 ) = (1, 1, 1) 7.3. Finitary combinatorial consequences and open questions 8. Khintchine-type recurrence for actions of semigroups 8.1. The group generated by a cancellative abelian semigroup 8.2. Notions of largeness 8.3. Extending main results to actions of cancellative abelian semigroups 8.4. Two combinatorial questions Appendix A. Proof of Lemma 3.6 References

Bohr sets in sumsets II: countable abelian groups

“…The Bohr sets in Proposition 10.1 and Theorem 1.7 have the same rank k and radius η. Proposition 10.1 gives k α −6 and η α 3 , where α = (δ 1 δ 2 r −1 ) 1/3 . If we are only interested in translates of Bohr sets (i.e., Bohr neighborhoods of some element), then better bounds are available.…”

“…To see this, assume ψ ∈ G does not have the form ψ • φ for some ψ ∈ H. Then there is a g ∈ ker φ such that ψ(g) = 1. 3 We then have…”

“…This proves the first statement in the claim, and the second statement is proved similarly. 3 Supposing ψ(g) = 1 for all g ∈ ker φ, we define a character ψ on H by ψ (φ(g)) = ψ(g). This is well defined, since φ(g) = φ(g ) implies ψ(g) = ψ(g ).…”

“…This point of view leads to a wider range of applications: we can consider linear maps on vector spaces and multiplication by an element in a ring (see Corollary 1.6 below). This broader perspective was also adopted in recent works [2,3] on Khintchine-type recurrence for actions of an abelian group. Third, we aim for uniformity in terms of rank and radius of the Bohr set in question, i.e., they are allowed to depend on d * (A) and other parameters, but not A itself.…”

Uniform syndeticity in multiple recurrence

“…In the main theorem of this paper, see Theorem 1.6 below, we establish the strong form of uniform syndeticity in Theorem 1.1 for an arbitrary (not necessarily countable) uniformly amenable group 1 . In particular, we strengthen the weak version of uniform syndeticity of Furstenberg-Katznelson in Theorem 1.2.…”