Title: Large intersections for multiple recurrence in abelian groups
Speaker: Ethan Ackelsberg (Ohio State University)
Speaker's URL: https://math.osu.edu/people/ackelsberg.1
Abstract: With the goal of a common extension of Khintchine’s recurrence theorem and Furstenberg’s multiple recurrence theorem in mind, Bergelson, Host, and Kra showed that, for any ergodic measure-preserving system (X, ℬ, μ, T), any measurable set A ∈ ℬ, and any ε > 0, there exist (syndetically many) n ∈ ℕ such that μ(A ∩ T^{n}A ∩ ... ∩ T^{kn}A) > μ(A)^{k+1} – ε if k ≤ 3, while the result fails for k ≥ 4. The phenomenon of large intersections for multiple recurrence was later extended to the context of ⊕𝔽p-actions by Bergelson, Tao, and Ziegler. In this talk, we will address and give a partial answer to the following question about large intersections for multiple recurrence in general abelian groups: given a countable abelian group G, what are necessary and sufficient conditions for a family of homomorphisms φ1, …, φk : G → G so that for any ergodic measure-preserving G-system (X, ℬ, μ, (Tg)g∈G), any A ∈ ℬ, and any ε > 0, there is a syndetic set of g ∈ G such that μ(A ∩ Tφ1(g)A ∩ ... ∩ Tφk(g)A) > μ(A)^{k+1} – ε? We will also discuss combinatorial applications in ℤ^{d} and (ℕ, ·). (Based on joint work with Vitaly Bergelson and Andrew Best and with Vitaly Bergelson and Or Shalom.)
Zoom: https://osu.zoom.us/j/94136097274
Meeting ID: 941 3609 7274
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