Arithmetic regularity lemmas and applications

Abstract

This thesis investigates various aspects of arithmetic regularity lemmas in the context of vector spaces over finite fields of prime characteristic. Chapter 1 obtains a generalisation of the induced arithmetic removal lemma of Bhattacharyya, Fischer, and Lovett [6] for translation-invariant arithmetic patterns, extending it to partition-regular patterns of complexity 1; this also strengthens the result of Fox, Tidor and Zhao [16] for general complexity-1 patterns. Chapter 2 establishes a wowzer-type lower bound on the size of the partition arising from the so-called strong arithmetic regularity lemma, which matches the bound of Conlon and Fox [12] for the analogous result in the graph-theoretic setting. The rest of the thesis concerns higher-order arithmetic regularity, in particular undertaking a study of local higher-order uniformity in Chapter 3. Two approaches to defining local uniformity on polynomial factors are proposed and subsequently applied to generalise two theorems of Green and Sanders [28] to polynomial factors of all degrees; specifically, it is shown that given any bounded function on a vector space over a field of characteristic 2, there is always a polynomial factor on whose zero atom the function is uniform, while over fields of characteristic greater than 2 this cannot be guaranteed. Finally, Chapters 4 and 5 address several questions concerning the quadratic arithmetic regularity lemmas of Terry and Wolf [60] under model-theoretically motivated tameness assumptions, including a proof of their conjecture that the set referred to as the quadratic Green-Sanders example has bounded VC₂-dimension.