Title: Combinatorial aspects of recurrence in Szemerédi’s theorem and applications addressing the interplay between additive and multiplicative largeness
Speaker: Daniel Glasscock (Ohio State University)
Abstract: Forty-two years ago, Endre Szemerédi proved that subsets of the natural numbers with positive upper density contain arbitrarily long arithmetic progressions. To this day, Szemerédi’s theorem and its relatives continue to stimulate new research in many fields, including combinatorics, additive number theory, harmonic analysis, and dynamics. In this talk, I will explain how some of the finer combinatorial aspects of recurrence in Szemerédi’s theorem can be derived from the (purely combinatorial) density Hales-Jewett theorem. I will then demonstrate how this extra information is useful in two applications relating notions of additive and multiplicative largeness.