Title: Large intersections for multiple recurrence in abelian groups
Speaker: Ethan Ackelsberg – Ohio State University
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 ∩ TnA ∩ … ∩ TknA) > 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 ℤd and (ℕ, ·). (Based on joint work with Vitaly Bergelson and Andrew Best and with Vitaly Bergelson and Or Shalom.)
Zoom link: https://osu.zoom.us/j/94136097274
Meeting ID: 941 3609 7274
Password: Mixing
Recorded Talk: https://osu.zoom.us/rec/share/TY64JIVXsqzNP_i1eNUIiwC0LriToGI6PVmOqPdJGnNuvNFRKkSLVvXiRP27RPU-.lyS_YtUQpBEuOhpC