Higher-order generalizations of stability and arithmetic regularity
Abstract
We define a natural notion of higher order stability and show that subsets of 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 , 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 -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.
Cite
@article{arxiv.2111.01739,
title = {Higher-order generalizations of stability and arithmetic regularity},
author = {C. Terry and J. Wolf},
journal= {arXiv preprint arXiv:2111.01739},
year = {2025}
}
Comments
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