An arithmetic algebraic regularity lemma

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 $H$.

💡 Research Summary

The paper establishes an “arithmetic regularity lemma” for groups definable in finite fields, extending Tao’s algebraic regularity lemma from graphs to a genuinely group‑theoretic setting. The authors work in the model‑theoretic framework of definable sets with bounded complexity (the length of the defining formula) and consider a definable group (G) together with a definable subset (D\subseteq G). The main result (Theorem 1.1, restated as Theorem 5.2) asserts that for any fixed complexity bound (M) there exists a constant (C=C(M)) such that one can find a normal definable subgroup (H\le G) of index at most (C) and of complexity at most (C) with the following property: for every pair of cosets (V,W) of (H) the bipartite graph ((V,W,E)) where (E={(v,w): vw^{-1}\in D}) is (C|{\mathbb F}|^{-1/2})-quasirandom. In other words, the edge distribution between any two cosets of (H) is essentially uniform, up to an error that decays as the square‑root of the field size.

A second, Fourier‑analytic formulation (Corollary 1.2 / Corollary 5.3) shows that the same subgroup (H) guarantees that for every (g\in G) and every non‑trivial irreducible representation (\pi) of (H), \