A Refinement of Cauchy-Schwarz Complexity, with Applications

Candela, Pablo; González-Sánchez, Diego; Szegedy, Balázs

doi:10.1007/978-3-030-83823-2_46

Cited by 3 publications

(4 citation statements)

References 8 publications

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“…Next, let us recall from (10) the definition of the group nilspaces H (p) i (for i ≥ 1), consisting of Z equipped with the filtration…”

Section: P Candela Et Almentioning

confidence: 99%

“…The proof of Theorem 1.3 occupies Section 2 and involves several steps, using in particular a recent refinement of the Generalized Von Neumann Theorem [10,11] (see also the recent work of Manners [32]). 1 Here Z n p is identified with [0, p − 1] n equipped with addition mod p, the usual way.…”

Section: Introductionmentioning

confidence: 99%

“…The proof of Theorem 1.3 occupies §2 and involves several steps, using, in particular, a recent refinement of the generalized Von Neumann theorem [10,11] (see also the recent work of Manners [32]).…”

Section: Introductionmentioning

confidence: 99%

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On higher-order Fourier analysis in characteristic p

Candela

González-Sánchez²,

Szegedy³

2023

Ergod. Th. Dynam. Sys.

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In this paper, the nilspace approach to higher-order Fourier analysis is developed in the setting of vector spaces over a prime field $\mathbb {F}_p$ , with applications mainly in ergodic theory. A key requisite for this development is to identify a class of nilspaces adequate for this setting. We introduce such a class, whose members we call p-homogeneous nilspaces. One of our main results characterizes these objects in terms of a simple algebraic property. We then prove various further results on these nilspaces, leading to a structure theorem describing every finite p-homogeneous nilspace as the image, under a nilspace fibration, of a member of a simple family of filtered finite abelian p-groups. The applications include a description of the Host–Kra factors of ergodic $\mathbb {F}_p^\omega $ -systems as p-homogeneous nilspace systems. This enables the analysis of these factors to be reduced to the study of such nilspace systems, with central questions on the factors thus becoming purely algebraic problems on finite nilspaces. We illustrate this approach by proving that for $k\leq p+1$ the kth Host–Kra factor is an Abramov system of order at most k, extending a result of Bergelson–Tao–Ziegler that holds for $k< p$ . We illustrate the utility of p-homogeneous nilspaces also by showing that the structure theorem yields a new proof of the Tao–Ziegler inverse theorem for Gowers norms on $\mathbb {F}_p^n$ .

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“…Next, let us recall from (10) the definition of the group nilspaces H (p) i (for i ≥ 1), consisting of Z equipped with the filtration…”

Section: P Candela Et Almentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

On higher-order Fourier analysis in characteristic p

Candela

González-Sánchez²,

Szegedy³

2023

Ergod. Th. Dynam. Sys.

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“…• work of Peluse [Pel19] on finding polynomial progressions in dense subsets of finite fields; • work of Peluse and Prendiville [PP19,PP20] and Peluse [Pel20] on cases of the polynomial Szemerédi theorem, which in particular required progress on a quantitative version of a "concatenation theorem" of Tao and Ziegler; • certain arguments in [Man18b] and [GM21] (and very recently [CGSS21]) related to the inverse theory of the Gowers norms.…”

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confidence: 99%

True complexity and iterated Cauchy--Schwarz

Manners¹

2021

Preprint

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We prove a polynomial bound in the "true complexity" problem of Gowers and Wolf. The proof uses only repeated applications of the Cauchy-Schwarz inequality, answering negatively a question posed by Gowers and Wolf.To choose and reason about the sequence of Cauchy-Schwarz steps needed, we need to introduce several layers of formalism and theory. The highest level of abstraction in this framework concerns building what we term "arithmetic circuits" encoding computations in multilinear algebra.It is plausible this machinery could be used to generate arithmetic inequalities in greater generality, and we state some conjectures along these lines. FREDDIE MANNERS 6. A proof of Theorem 1.1.5 80 6.1. The structure of the proof 80 6.2. Some stashing-type proofs 81 6.3. The Bridge gate 83 6.4. The super AGate 84 6.5. The big AggreGate 86 6.6. StarGates and Lemma 6.1.2 88 6.7. Building a StarGate 91 6.8. Assigning a StarGate 94 7. Conjectures 99 7.1. Cases to avoid 99 7.2. A precise phrasing 100 7.3. Requiring an elementary argument 101 7.4. A special case 102 References 102 1. Note we have control of Λ Φ by some Gowers norm • U scs+1 , by the Cauchy-Schwarz complexity argument [GW10, Theorem 2.3]. Hence we are free to modify f i by small errors in the • U scs +1 -norm. 2. Apply an inverse theorem for the Gowers •GTZ12], or for quantitative bounds [GM21, Man18b]). 3. Taking steps 1 and 2 together, we may assume WLOG that f i are "U scs+1 structured functions": i.e., nilsequences (if n = 1 and p is large) or phase polynomials (if p is fixed and n is large).11 To be accurate, some of these results proved the original non-multilinear conjecture, with However, Hatami and Lovett [HL11] (in finite fields) and Altman's argument (for n = 1) prove the multilinear version discussed here.12 The original version of [GT10a] did not mention this extra hypothesis, but it was later observed by Altman that the proofs assumed it implicitly.

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