Stable arithmetic regularity in the finite field model

Abstract

The arithmetic regularity lemma for Fpn, proved by Green in 2005, states that given a subset A⊆double-struckFpn, there exists a subspace H⩽double-struckFpn of bounded codimension such that A is Fourier‐uniform with respect to almost all cosets of H. It is known that in general, the growth of the codimension of H is required to be of tower type depending on the degree of uniformity, and that one must allow for a small number of non‐uniform cosets. Our main result is that, under a natural model‐theoretic assumption of stability, the tower‐type bound and non‐uniform cosets in the arithmetic regularity lemma are not necessary. Specifically, we say that a set A⊆double-struckFpn is k‐stable if there are no a1,…,ak,b1,…,bk∈double-struckFpn such that ai+bj∈A if and only if i⩽j. We prove an arithmetic regularity lemma for k‐stable subsets A⊆double-struckFpn in which the bound on the codimension of the subspace is a polynomial (depending on k) in the degree of uniformity, and in which there are no non‐uniform cosets. This result is an arithmetic analogue of the stable graph regularity lemma proved by Malliaris and Shelah.

Quantitative structure of stable sets in finite abelian groups

“…The proof of Theorem 2 in [5] relies entirely on model-theoretic techniques, specifically the "localisation" of existing results by Hrushovsky and Pillay [11] in stable group theory. The authors also obtained a stronger structural result than that proved in [19] (see the discussion in Section 5). In a separate paper [6], Conant, Pillay and Terry also proved a result for subsets of finite groups whose left-translates have bounded VC-dimension.…”

“…Motivated by model-theoretic considerations and the graph-theoretic result [12], the authors proved an arithmetic regularity lemma for stable subsets of high-dimensional vector spaces over a fixed finite field of prime order in [19]. In order to be able to state it, we recall the definition of a k-stable set from [19]. Throughout this paper, G will be a general finite abelian group unless explicitly stated otherwise.…”

“…For a discussion of the significance of the arithmetic regularity in arithmetic combinatorics, and the relationship between Theorem 1 and Green's arithmetic regularity lemma [9], we refer the reader to the introduction of [19].…”

“…Shortly after the preprint [19] appeared, the first author, in collaboration with Conant and Pillay [5], obtained the following qualitative arithmetic regularity lemma for stable subsets of general finite groups.…”

“…These are entirely standard in arithmetic combinatorics, allowing many arguments in the toy setting of F n p to be transferred to general finite abelian groups via a process known as Bourgainisation. However, a key part of the argument in [19] does not yield to generalisation in this way: it does not seem possible to show that a pair of Bohr sets gives rise to a sufficiently "regular pair" in the relevant Cayley graph [19,Lemma 7].…”