A quantitative inverse theorem for the $U^4$ norm over finite fields

Gowers, W. T.; Milićević, Luka

doi:10.48550/arxiv.1712.00241

Cited by 5 publications

(11 citation statements)

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“…Although no reduction of the exact shape of (ii) appears in the literature, this kind of correspondence is closely related to aspects of previous works [GT08,Gow01,GM17], and is anticipated explicitly in [KZ18,KZ17b]. The proof here builds on the previous arguments.…”

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confidence: 68%

“…Recently, Gowers and Milićević [GM17] obtained a quantitative result in this setting for s = 3, where the relevant parameters (phrased rather differently to here, but corresponding primarily to the parameter ε −1 above) are given a doubleexponential bound. There are many points of comparison between this work and [GM17]; however, it is not straightforward to make their methods work in the cyclic setting (even for s = 3), and not straightforward to make the methods of this paper work over finite fields, so these results are related but nonetheless disjoint.…”

mentioning

confidence: 97%

“…That said, proving an analogue of Theorem 1.3.3 in this finite field setting is a primary motivation for the ongoing programme [KZ18,KZ17a,KZ17b], making this work closely related, although the issues that arise in the finite field setting often have a different flavour. Similarly, [GM17] treats a related bilinear problem, again over finite fields.…”

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confidence: 99%

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Quantitative bounds in the inverse theorem for the Gowers $U^{s+1}$-norms over cyclic groups

Manners¹

2018

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We provide a new proof of the inverse theorem for the Gowers U s+1 -norm over groups H = Z/N Z for N prime. This proof gives reasonable quantitative bounds (the worst parameters are double-exponential), and in particular does not make use of regularity or non-standard analysis, both of which are new for s ≥ 3 in this setting. 2 FREDDIE MANNERS Appendix C. Algebraic definitions and facts concerning nilmanifolds and related objects 111 C.1. Filtered groups and Host-Kra cubes 112 C.2. Nilmanifolds and complexity 112 C.3. Polynomial maps 114 C.4. Metrics 119 C.5. Miscellaneous results 121 References 122

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confidence: 68%

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confidence: 97%

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Quantitative bounds in the inverse theorem for the Gowers $U^{s+1}$-norms over cyclic groups

Manners¹

2018

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“…The best bounds, which are quasipolynomial in nature, are due to Sanders [61]. Until recent work of Gowers and Milićević [32], a corresponding result for the U 4 norm on F n p was only known in a qualitative fashion [7,70]. Special considerations apply in small characteristic [52,71].…”

Section: 2mentioning

confidence: 99%

Higher-order generalizations of stability and arithmetic regularity

Terry¹,

Wolf²

2021

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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...

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“…The original proofs of these inverse theorems give extremely bad or even ineffective quantitative bounds. Recently Manners proved quantitative bounds for the U k+1 -inverse theorem over Z/N Z [14] and Gowers and Milićević proved quantitative bounds for the U k+1 -inverse theorem over F n p when p > k [6,5].…”

Section: Introductionmentioning

confidence: 99%

Quantitative bounds for the $U^4$-inverse theorem over low characteristic finite fields

Tidor¹

2021

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This paper gives the first quantitative bounds for the inverse theorem for the Gowers U 4 -norm over F n p when p = 2, 3. We build upon earlier work of Gowers and Milićević who solved the corresponding problem for p ≥ 5. Our proof has two main steps: symmetrization and integration of low-characteristic trilinear forms. We are able to solve the integration problem for all k-linear forms, but the symmetrization problem we are only able to solve for trilinear forms. We pose several open problems about symmetrization of low-characteristic k-linear forms whose resolution, combined with recent work of Gowers and Milićević, would give quantitative bounds for the inverse theorem for the Gowers U k+1 -norm over F n p for all k, p.

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A quantitative inverse theorem for the $U^4$ norm over finite fields

Cited by 5 publications

References 13 publications

Quantitative bounds in the inverse theorem for the Gowers $U^{s+1}$-norms over cyclic groups

Quantitative bounds in the inverse theorem for the Gowers $U^{s+1}$-norms over cyclic groups

Higher-order generalizations of stability and arithmetic regularity

Quantitative bounds for the $U^4$-inverse theorem over low characteristic finite fields

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