**Title:** Multiple recurrence for two commuting transformations

**Speaker:** Qing Chu

**Speaker Info:** University of Marne la Vallee

**Brief Description:**

**Special Note**:

**Abstract:**

A quantitative version of PoincarĂ©'s Recurrence Theorem is Khintchine's Recurrence Theorem, which states that if A is a set of positive measure, then for every epsilon>0, the set {n: \mu(A\cap T^{-n} A)>\mu(A)^2-\epsilon\} has bounded gaps.Furstenberg proved a Multiple Recurrence Theorem, soon after, he and Katznelson generalized this result to commuting transformations. One question is whether there exist an analogous quantitative version of Furstenberg's Theorem, and furthermore a quantitative version of Furstenberg and Katznelson's Theorem. Bergelson, Host and Kra gave a comprehensive answer to the first question.

In this talk, we focus on the systems endowed with two commuting transformations. We show a result similar but not identical to that of Bergelson, Host and Kra. One important tool for the proof is the machinery of "magic systems" established by B. Host, we discuss how it is used in the proof.

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