Combinatorial and dynamical properties of polynomial progressions

Abstract

In this thesis, we study combinatorial and dynamical questions on polynomial progressions that are connected with the polynomial Szemerédi theorem of Bergelson and Leibman. Some of these questions stem from additive combinatorics whereas others have ergodic-theoretic flavour. Firstly, we prove upper bounds for the size of subsets of finite fields lacking certain polynomial progressions. Specifically, we look at two single-dimensional families of progressions and one multidimensional family. In doing so, we obtain asymptotics for the number of certain progressions in subsets of finite fields with quantitative error terms; we also get a quantitative control of certain polynomial configurations by low-degree Gowers norms. These results are obtained using discrete Fourier analysis, a basic theory of Gowers norms, the degree-lowering argument of Peluse, and variations of the PET induction scheme of Bergelson and Leibman. Secondly, we qualitatively study several notions of complexity of polynomial progressions. One of them comes from additive combinatorics and describes the smallest-degree Gowers norm controlling a given configuration. Another two originate in ergodic theory and refer to the smallest characteristic factor for the convergence of the multiple ergodic averages associated with the progression. The last one is purely algebraic, concerning the algebraic relations between terms of the progression. We conjecture that these four notions agree for all polynomial progressions. We show this for all homogeneous progressions, a large class of progressions that includes most of the progressions for which complexity results have previously been obtained, and many more. We also prove a number of smaller results: the equivalence of true and algebraic complexity for a certain family of inhomogeneous polynomial progressions, asymptotics for the count of progressions of complexity 1 or multiple recurrence results for these configurations. Our proofs apply techniques from higher order Fourier analysis and ergodic theory. In the process of deriving our results, we give new equidistribution results on nilmanifolds for certain types of polynomial sequences.

Date of Award	31 Dec 2021
Original language	English
Awarding Institution	The University of Manchester
Supervisor	Donald Robertson (Supervisor) & Yuri Bazlov (Supervisor)