**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 ∩ 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 ⊕𝔽_{p}-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 φ_{1}, …, φ_{k} : G → G so that for any ergodic measure-preserving G-system (X, ℬ, μ, (T_{g})_{g}_{∈}_{G}), any A ∈ ℬ, and any ε > 0, there is a syndetic set of g ∈ G such that μ(A ∩ T_{φ1(g)}A ∩ … ∩ T_{φk(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 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