An arithmetic algebraic regularity lemma

TL;DR

This paper establishes an arithmetic regularity lemma for groups definable in finite fields, showing that such groups can be decomposed into parts with quasirandom properties, extending Tao's algebraic regularity lemma to an arithmetic setting.

Contribution

It introduces an arithmetic regularity lemma for definable groups in finite fields, providing a structural decomposition with quasirandomness properties analogous to graph regularity lemmas.

Findings

Existence of a normal definable subgroup with bounded complexity and index.

Bipartite graphs between cosets are quasirandom with bounds depending on the field size.

Fourier coefficients of intersections with cosets are bounded by a term depending on the field size.

Abstract

We give an 'arithmetic regularity lemma' for groups definable in finite fields, analogous to Tao's 'algebraic regularity lemma' for graphs definable in finite fields. More specifically, we show that, for any $M>0$, any finite field $\mathbf{F}$, and any definable group $(G,\cdot)$ in $\mathbf{F}$ and definable subset $D\subseteq G$, each of complexity at most $M$, there is a normal definable subgroup $H\leqslant G$, of index and complexity $O_M(1)$, such that the following holds: for any cosets $V,W$ of $H$, the bipartite graph $(V,W,xy^{-1}\in D)$ is $O_M(|\mathbf{F}|^{-1/2})$-quasirandom. Various analogous regularity conditions follow; for example, for any $g\in G$, the Fourier coefficient $||\widehat{1}_{H\cap Dg}(\pi)||_{\mathrm{op}}$ is $O_M(|\mathbf{F}|^{-1/8})$ for every non-trivial irreducible representation $\pi$ of $H$.