Higher-order generalizations of stability and arithmetic regularity

Abstract

We define a natural notion of higher order stability and show that subsets of ${\mathbb{F}}_p^n$ that are tame in this sense can be approximately described by a union of low-complexity quadratic varieties, up to linear error. This generalizes the arithmetic regularity lemma for stable subsets of ${\mathbb{F}}_p^n$, proved in earlier work of the authors, to the realm of higher-order Fourier analysis. This result is strictly stronger than the structure theorem for sets of bounded ${\mathrm{VC}}_2$-dimension, first proved by the authors in earlier versions of this paper and now available as a separate manuscript arXiv:2510.12867. Taken together, these results provide group theoretic analogues of results obtained for 3-uniform hypergraphs in arXiv:2111.01737.

110 pages. The proof of Theorem 1.12 has been transferred and completely reworked into the separate manuscript arXiv:2510.12867 to allow the present version of this paper to focus exclusively on higher-order stability. Cross references updated