Stable arithmetic regularity in the finite field model
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Publication Date
2019-02Journal Title
Bulletin of the London Mathematical Society
ISSN
0024-6093
Publisher
Wiley
Volume
51
Issue
1
Pages
70-88
Type
Article
This Version
AM
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Terry, C., & Wolf, J. (2019). Stable arithmetic regularity in the finite field model. Bulletin of the London Mathematical Society, 51 (1), 70-88. https://doi.org/10.1112/blms.12211
Abstract
The arithmetic regularity lemma for $\mathbb{F}_p^n$, proved by Green in
2005, states that given a subset $A\subseteq \mathbb{F}_p^n$, there exists a
subspace $H\leq \mathbb{F}_p^n$ 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 prove an arithmetic
regularity lemma for $k$-stable subsets $A\subseteq \mathbb{F}_p^n$ 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.
Identifiers
External DOI: https://doi.org/10.1112/blms.12211
This record's URL: https://www.repository.cam.ac.uk/handle/1810/286537
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