Seminar 11.2.17 Johnson

Title: Khintchine recurrence for upper Banach density along filters

SpeakerJohn Johnson (Ohio State University)

Abstract: Khintchine’s recurrence theorem is a well-known extension of Poincaré’s recurrence theorem improving the “quality” of the recurrence in a measure-preserving transformation on a probability space: if $A$ is a subset of the probability space with positive measure, then the collection $\{ n \in \mathbb{N} : \mu(A \cap T^{-n}A) > 0 \}$ has bounded gaps. This recurrence theorem also has a combinatorial analog for sets with positive upper Banach density: if $A \subseteq \mathbb{N}$ has positive upper Banach density $d^*(A) > 0$, then the collection $\{ n \in \mathbb{N} : d^ *(A \cap -n + A) > 0 \}$ has bounded gaps. By iteration, this combinatorial theorem yields Szemerédi’s affine cube lemma as an easy corollary. (The affine cube lemma is one important component in the combinatorial proof of Roth’s theorem on three-term arithmetic progressions.)

In joint work with Florian Richter, we define a notion of upper Banach density, relative to filters on the positive integers, and, under some relatively mild conditions on the underlying filters, prove an analog of Khintchine’s recurrence theorem for this generalization of Banach density as well. As a combinatorial consequence, we can obtain a restricted version of Szemerédi’s affine cube lemma