Seminar 2.6.20 Ferre Moragues

Title: Combinatorial notions of largeness and their interaction with Ergodic Theory

Speaker: Andreu Ferre Moragues – The Ohio State University

Abstract: A theorem due to Hindman states that for any set E⊆ℕ withd∗(E):=limsupN−M→∞|E∩{M,…,N−1}|/(N−M) >0, and any ε>0 there exists some N∈ℕ such that d∗(⋃N i=0(E−i))>1−ε. Hindman’s theorem, a guiding theme for the talk, will allow us to distinguish between two notions of largeness: upper density (d¯) and upper Banach density (d∗).

We will also see how Hindman’s theorem allows for a deeper understanding of Furstenberg’s correspondence principle. Indeed, one can show that an appropriate version of Furstenberg’s correspondence principle yields a dynamical proof of this theorem which is simpler than the original combinatorial one and can be generalized to amenable semigroups.

Moreover, a general version of Hindman’s theorem helps characterize WM groups (i.e., groups with the property that any ergodic measure preserving action (Tg)g∈G on a probability space (X,B,μ) is weakly mixing). Time permitting, we will discuss the strategy of the proofs and how far the results can be extended. The talk is based on a joint work with Dr. Bergelson.