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Higher-order generalizations of stability and arithmetic regularity

C. Terry, J. Wolf

TL;DR

The paper develops a higher-order analogue of stability for subsets of elementary abelian p-groups by introducing NFOP_2, a ternary stability-like notion that aligns with quadratic Fourier analysis. It proves a quadratic arithmetic regularity lemma: NFOP_2 subsets of $\mathbb{F}_p^n$ can be described up to linear error by unions of atoms of a high-rank quadratic factor of bounded complexity, with irregular atoms concentrated in a small linear subset, reflecting a refined structural description beyond VC-dimension bounds. The work systematically builds tools—linear and quadratic factors, counting lemmas, reduced structures, and tree-encoding methods—to connect order properties, VC_2-dimension, and higher-arity tameness, and it contrasts NFOP_2 with NIP_2 via quadratic Green-Sanders-type examples that demonstrate sharpness. It also relates the arithmetic results to hypergraph regularity, showing parallel phenomena in 3-uniform hypergraphs and outlining directions for generalizing to higher arity, other groups, and model-theoretic nilstructures. The combination of quadratic regularity, reduced-structure analysis, and tree-encoding techniques provides a blueprint for understanding higher-order stability in additive combinatorics and its interactions with hypergraph theory and model theory.

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.

Higher-order generalizations of stability and arithmetic regularity

TL;DR

The paper develops a higher-order analogue of stability for subsets of elementary abelian p-groups by introducing NFOP_2, a ternary stability-like notion that aligns with quadratic Fourier analysis. It proves a quadratic arithmetic regularity lemma: NFOP_2 subsets of can be described up to linear error by unions of atoms of a high-rank quadratic factor of bounded complexity, with irregular atoms concentrated in a small linear subset, reflecting a refined structural description beyond VC-dimension bounds. The work systematically builds tools—linear and quadratic factors, counting lemmas, reduced structures, and tree-encoding methods—to connect order properties, VC_2-dimension, and higher-arity tameness, and it contrasts NFOP_2 with NIP_2 via quadratic Green-Sanders-type examples that demonstrate sharpness. It also relates the arithmetic results to hypergraph regularity, showing parallel phenomena in 3-uniform hypergraphs and outlining directions for generalizing to higher arity, other groups, and model-theoretic nilstructures. The combination of quadratic regularity, reduced-structure analysis, and tree-encoding techniques provides a blueprint for understanding higher-order stability in additive combinatorics and its interactions with hypergraph theory and model theory.

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.

Paper Structure

This paper contains 35 sections, 84 theorems, 272 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Stability and graph regularity
  3. Stable group theory and linear arithmetic regularity
  4. The search for higher arity stability
  5. Formal introduction to tame linear arithmetic regularity
  6. Main results
  7. Connections to work on hypergraphs
  8. Future directions
  9. Preliminaries
  10. Notation
  11. Linear and quadratic factors
  12. Basic properties of $\mathrm{FOP}_2$ and important examples
  13. Relationships between $\mathrm{FOP}_2$ and other notions of tameness
  14. Properties of quadratic atoms
  15. The linear Green-Sanders example
  16. ...and 20 more sections

Key Result

Theorem 1.2

For all primes $p$, integers $k\geq 1$ and $\epsilon\in (0,1)$, there exists $M=\exp(\epsilon^{-O_{p,k}(1)})$ such that for all sufficiently large $n$, the following holds. For any $k$-stable $A\subseteq G= \mathbb{F}_p^n$, there exists $H\leqslant G$ of index at most $M$ such that

Figures (3)

  • Figure :
  • Figure :
  • Figure :

Theorems & Definitions (200)

  • Definition 1.1: Order property and stability for subsets of groups
  • Theorem 1.2: Arithmetic regularity lemma for stable sets Terry.2019
  • Definition 1.3: $\mathrm{VC}$-dimension of a subset of a group
  • Theorem 1.4: Arithmetic regularity lemma for sets of bounded $\mathrm{VC}$-dimension Alon.2018is
  • Remark 1.5
  • Definition 1.6: Green-Sanders example
  • Proposition 1.7: Green-Sanders example has bounded $\mathrm{VC}$-dimension
  • Proposition 1.8: Green-Sanders example requires an irregular coset
  • Theorem 1.9: Strong arithmetic regularity for stable sets Conant.2017p4
  • Theorem 1.10: Strong arithmetic regularity for sets of bounded $\mathrm{VC}$-dimension Conant.2018zd
  • ...and 190 more