Linear Forms and Quadratic Uniformity for Functions On

Gowers, Timothy; Wolf, Julia

doi:10.1112/s0025579311001264

Cited by 27 publications

(44 citation statements)

References 12 publications

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“…Note that in [GW10a] the authors had to work much harder to obtain a bound on the number of terms in the decomposition, rather than just the 1 norm of its coefficients. Our decomposition approach gives such a bound immediately and is equivalent from a quantitative point of view: we can bound the number of terms here by 1/η 2 , which is exponential in 1/ε.…”

Section: Overview Of Results and Techniquesmentioning

confidence: 99%

“…We note that a third term with bounded 1 norm also appears in the (non-constructive) decompositions obtained in [GW10a].…”

Section: From Decompositions To Correlation Testingmentioning

confidence: 99%

“…However, unlike the decomposition into Fourier characters, the decomposition in terms of quadratic phases is not necessarily unique, as the quadratic phases do not form a basis for the space of functions on F n 2 . Quadratic decomposition theorems have found several number-theoretic applications, notably in a series of papers by Gowers and the second author [GW10c,GW10a,GW10b], as well as [Can10] and [HL11].…”

Section: Introductionmentioning

confidence: 99%

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Quadratic Goldreich--Levin Theorems

Tulsiani¹,

Wolf²

2014

SIAM J. Comput.

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Decomposition theorems in classical Fourier analysis enable us to express a bounded function in terms of few linear phases with large Fourier coefficients plus a part that is pseudorandom with respect to linear phases. The Goldreich-Levin algorithm [GL89] can be viewed as an algorithmic analogue of such a decomposition as it gives a way to efficiently find the linear phases associated with large Fourier coefficients.In the study of "quadratic Fourier analysis", higher-degree analogues of such decompositions have been developed in which the pseudorandomness property is stronger but the structured part correspondingly weaker. For example, it has previously been shown that it is possible to express a bounded function as a sum of a few quadratic phases plus a part that is small in the U 3 norm, defined by Gowers for the purpose of counting arithmetic progressions of length 4. We give a polynomial time algorithm for computing such a decomposition.A key part of the algorithm is a local self-correction procedure for Reed-Muller codes of order 2 (over F n 2 ) for a function at distance 1/2−ε from a codeword. Given a function f : F n 2 → {−1, 1} at fractional Hamming distance 1/2 − ε from a quadratic phase (which is a codeword of ReedMuller code of order 2), we give an algorithm that runs in time polynomial in n and finds a codeword at distance at most 1/2 − η for η = η(ε). This is an algorithmic analogue of Samorodnitsky's result [Sam07], which gave a tester for the above problem. To our knowledge, it represents the first instance of a correction procedure for any class of codes, beyond the list-decoding radius.In the process, we give algorithmic versions of results from additive combinatorics used in Samorodnitsky's proof and a refined version of the inverse theorem for the Gowers U 3 norm over F n 2 .

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Section: Overview Of Results and Techniquesmentioning

confidence: 99%

“…We note that a third term with bounded 1 norm also appears in the (non-constructive) decompositions obtained in [GW10a].…”

Section: From Decompositions To Correlation Testingmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Quadratic Goldreich--Levin Theorems

Tulsiani¹,

Wolf²

2014

SIAM J. Comput.

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“…One way of proving this, due to Green and Tao [27], is modelled on arguments from ergodic theory: if f is large one uses the inverse theorem to find a function F ∈ F that correlates well with f , defines a "sigma-algebra" with respect to which F is "approximately measurable", and then repeats the process with f −P f, continuing until the desired decomposition is achieved. Another approach uses the Hahn-Banach theorem to obtain a contradiction of the inverse theorem if the desired decomposition does not exist: see [23,24] for some applications of this idea, and [21] for a general discussion of the method.…”

Section: W T Gowersmentioning

confidence: 99%

Untitled

2017

Bull. Amer. Math. Soc.

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Abstract. This is a survey of the use of Fourier analysis in additive combinatorics, with a particular focus on situations where it cannot be straightforwardly applied but needs to be generalized first. Sometimes very satisfactory generalizations exist, while sometimes we have to make do with theories that have some of the desirable properties of Fourier analysis but not all of them. In the latter case, there are intriguing hints that there may be more satisfactory theories yet to be discovered. This article grew out of the Colloquium Lectures at the Joint Meeting of the AMS and the MAA, given in Seattle in January 2016.

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“…To the best of my knowledge, there are two types of inverse theorems in additive combinatorics, namely the inverse sumset theorems of Freiman type (see, e.g., [78,117,118,152,155,275,299,306,307,308,318] and [116,238]), and inverse theorems for the Gowers norms (see, e.g., [143,146,147,148,153,162,156,159,163,164,176,181,189,203,224,225,233,270,326,332]). It is interesting that the inverse conjecture leads to a finite field version of Szemerédi's theorem [320]: Let F p be a finite field.…”

Section: Szemerédi's and Green-tao Theorems And Their Generalizationsmentioning

confidence: 99%

Additive Combinatorics: With a View Towards Computer Science and Cryptography—An Exposition

Bibak

2013

Springer Proceedings in Mathematics &Amp; Statistics

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Recently, additive combinatorics has blossomed into a vibrant area in mathematical sciences. But it seems to be a difficult area to define -perhaps because of a blend of ideas and techniques from several seemingly unrelated contexts which are used there. One might say that additive combinatorics is a branch of mathematics concerning the study of combinatorial properties of algebraic objects, for instance, Abelian groups, rings, or fields. This emerging field has seen tremendous advances over the last few years, and has recently become a focus of attention among both mathematicians and computer scientists. This fascinating area has been enriched by its formidable links to combinatorics, number theory, harmonic analysis, ergodic theory, and some other branches; all deeply cross-fertilize each other, holding great promise for all of them! In this exposition, we attempt to provide an overview of some breakthroughs in this field, together with a number of seminal applications to sundry parts of mathematics and some other disciplines, with emphasis on computer science and cryptography.

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Linear Forms and Quadratic Uniformity for Functions On

Cited by 27 publications

References 12 publications

Quadratic Goldreich--Levin Theorems

Quadratic Goldreich--Levin Theorems

Untitled

Additive Combinatorics: With a View Towards Computer Science and Cryptography—An Exposition

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