Higher-Order Generalizations of Stability and Arithmetic Regularity

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 memoir and now available as a separate article. Taken together, these results provide group-theoretic analogs of results obtained by the authors for 3-uniform hypergraphs.