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 $H⩽G$, of index and complexity $O_M(1)$, such that the following holds: for any cosets $V,W$ of $H$, the bipartite graph $(V,W,x{y}^{-1}\in D)$ is $O_M(\mathrm{\mid }\mathbf{F}{\mathrm{\mid }}^{-1\mathrm{/}2})$-quasirandom. Various analogous regularity conditions follow; for example, for any $g\in G$, the Fourier coefficient $\mathrm{\mid }\mathrm{\mid }{\widehat{1}}_{H\cap Dg}(\pi )\mathrm{\mid }{\mathrm{\mid }}_{\mathrm{o}\mathrm{p}}$ is $O_M(\mathrm{\mid }\mathbf{F}{\mathrm{\mid }}^{-1\mathrm{/}8})$ for every non-trivial irreducible representation $\pi$ of $H$.