Stable arithmetic regularity in the finite field model

Terry, C; Wolf, J

Stable arithmetic regularity in the finite field model

Authors

Terry, C

Wolf, J

Publisher

Wiley
https://doi.org/10.1112/blms.12211

Description

The arithmetic regularity lemma for Fpn, proved by Green in 2005, states that given a subset A⊆Fpn, there exists a subspace H⩽Fpn 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⊆Fpn is k‐stable if there are no a1,…,ak,b1,…,bk∈Fpn such that ai+bj∈A if and only if i⩽j. We prove an arithmetic regularity lemma for k‐stable subsets A⊆Fpn 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.