An arithmetic algebraic regularity lemma
Anand Pillay, Atticus Stonestrom
TL;DR
The paper develops an arithmetic regularity lemma for groups definable in finite fields, establishing the existence of a bounded-index normal definable subgroup that induces strong quasirandomness for the associated definable bipartite graphs. It blends model-theoretic notions (simplicity, S1, pseudofinite fields) with non-abelian Fourier analysis to obtain spectral control and uniform density bounds for definable sets, culminating in a Schreier-type regularity statement with $O_M(|\mathbf{F}|^{-1/2})$-quasirandomness. A sharper bound is then achieved, giving $O(|\mathbf{F}|^{-1/2})$-quasirandomness under the same hypotheses, and the results are specialized to algebraic groups, including additive and Heisenberg examples, as well as semisimple groups via universal covers. The work connects to Green’s finite-field model and extends Tao’s algebraic regularity framework to an arithmetic context, with concrete consequences for definable subsets of various groups in large characteristic.
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 $||\widehat{1}_{H\cap Dg}(π)||_{\mathrm{op}}$ is $O_M(|\mathbf{F}|^{-1/8})$ for every non-trivial irreducible representation $π$ of $H$.