Higher-order generalizations of stability and arithmetic regularity

Abstract

We define a natural notion of higher-order stability and show that subsets of F n p that are tame in this sense can be approximately described by a union of low-complexity quadratic subvarieties up to linear error. This generalizes the arithmetic regularity lemma for stable subsets of F n p proved by the authors in [73], and subsequent refinements and generalizations [19,74], to the realm of higher-order Fourier analysis. HIGHER-ORDER GENERALIZATIONS OF STABILITY AND ARITHMETIC REGULARITY 5.2. Translation to trees 5.3. Proof of main NFOP 2 theorems 5.4. A preliminary generalized stable regularity lemma 5.5. Iterating to obtain a stronger version Appendix A. Hereditary properties A.1. Equivalent definitions A.2. Closeness A.3. An unstable property which is close to stable Appendix B. Boolean combinations of quadratic atoms are NHOP 2 Appendix C. Properties of the linear Green-Sanders example C.1. No 4-IP in GS(p, n) C.2. No 4-HOP 2 in GS(3, n) Appendix D. Density lemmas Appendix E. Counting lemmas and corollaries ReferencesThis theorem is analogous to the structural results for sets of bounded VC-dimension found in [3,22,65,73]. As an immediate corollary, we have the following.Corollary 1.19 (NIP 2 implies AQA). Let p ≥ 3 be a prime and let P be an elementary p-group property which is NIP 2 . Then P is almost quadratically atomic.We prove an analogous result in the hypergraph setting in [76] (see also [15]). We also obtain the following structural result for sets of bounded VC 2 -dimension.Corollary 1.20 (Structure of sets with bounded VC 2 -dimension). For all ǫ > 0, k ≥ 1, and rank functions ρ : N → R + , there are integers D, N such that the following holds. For all n ≥ N and A ⊆ F n p with VC 2 (A) < k, there is a quadratic factor B of complexity (ℓ, q) and rank at least ρ(ℓ+q), and a set I ⊆ At(B) such that ℓ, q ≤ D, andCorollary 1.20 is a higher-order analogue of earlier results [3,22,65,73] that state that a set of bounded VC-dimension is approximately a union of cosets of a subgroup of bounded index (in other words, a union of atoms from a purely linear factor).In fact, we believe that something stronger is true. In analogy with Definition 1.3, we say that P is almost strongly quadratically atomic if for all ǫ > 0, all functions g : N → N, and all rank functions ρ : N → R + , there are integers D, N such that for all n ≥ N, and all (G, A) ∈ P with |G| ≥ p n , the following holds. There exists (ℓ, q) satisfying ℓ + q ≤ D, and a quadratic factor B in G of complexity (ℓ, q) and rank at least ρ(ℓ + q), which is ǫ-almost ǫp −g(ℓ+q) -atomic with respect to A.Conjecture 1.21. If an epGP is NIP 2 , then it is almost strongly quadratically atomic.As was the case for linear decompositions (see Proposition 1.10), it is not difficult to show that whether or not a group property satisfies any part of Definition 1.15 depends only on its ∼-class, see Appendix A.2.Proposition 1.22. If P and P ′ are close then P is (strongly) quadratically atomic/almost (strongly) quadratically atomic if and only if P ′ is.In light of this...