Stable arithmetic regularity in the finite field model
Bulletin of the London Mathematical Society
The arithmetic regularity lemma for F_p^n, proved by Green in 2005, states that given a subset A⊆F_p^n, there exists a subspace H≤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
... f 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⊆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.