Othmane Rih: Additive Structures in the Integers

Abstract

In this talk, I’ll introduce combinatorial Ramsey theory, a beautiful area of mathematics that studies patterns that must appear when we divide numbers or sets into parts. Classic examples include finding arithmetic progressions inside dense sets of integers. I’ll then explain how ideas from ergodic theory, a branch of dynamics, provide powerful tools to solve these kinds of problems in new ways. Finally, I’ll talk about higher-order Fourier analysis and how it helps us understand complex patterns in number sets, touching on famous results like the Gowers’ uniformity theory, and the Green–Tao theorem about arithmetic progressions in the primes.