Higher Order Fourier Analysis

Tao, Terence

doi:10.1090/gsm/142

Cited by 97 publications

(104 citation statements)

References 78 publications

(31 reference statements)

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“…Proof : Property 1) follows directly from (5). To see 2), note that d dt xv, wy s " xv t , wy s`x v, w t y s " xP v, wy s`x v, P wy s " trpP qxv, wy s " 0 (by direct calculation).…”

Section: A Nonlinear Fourier Coefficientsmentioning

confidence: 93%

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Integrable communication channels and the nonlinear fourier transform

Yousefi

Kschischang

2013

2013 IEEE International Symposium on Information Theory

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This paper considers the transmission of information over integrable channels, a class of (mainly nonlinear) channels described by a Lax operator-pair. For such channels, the nonlinear Fourier transform, a powerful tool in soliton theory and exactly solvable models, plays the same role in "diagonalizing" the channel that the ordinary Fourier transform plays for linear convolutional channels. A transmission strategy encoding information in the nonlinear Fourier spectrum, termed nonlinear frequency-division multiplexing, is proposed for integrable channels that is the nonlinear analogue of orthogonal frequencydivision multiplexing commonly used in linear channels. A central and motivating example is fiber-optic data transmission, for which the proposed transmission technique deals with both dispersion and nonlinearity directly and unconditionally without the need for dispersion or nonlinearity compensation methods.

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“…Proof : Property 1) follows directly from (5). To see 2), note that d dt xv, wy s " xv t , wy s`x v, w t y s " xP v, wy s`x v, P wy s " trpP qxv, wy s " 0 (by direct calculation).…”

Section: A Nonlinear Fourier Coefficientsmentioning

confidence: 93%

“…Definition 4 (Nonlinear Fourier Transform [5], [6]). Let qptq be a sufficiently smooth function in L 1 pRq.…”

Section: B the Nonlinear Fourier Transformmentioning

confidence: 99%

Integrable communication channels and the nonlinear fourier transform

Yousefi

Kschischang

2013

2013 IEEE International Symposium on Information Theory

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“…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. Suppose that δ > 0, and A ⊂ F n p with |A| ≥ δ|F n p |.…”

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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“…The uniformity norms

{false∥ \cdot false∥}_{U^{s} [N]}

were first introduced by Gowers in his work on a (higher order) Fourier‐analytic proof of Szemerédi's Theorem . Here, we just give the basic facts; for an extensive background, we refer to or .…”

Section: Introductionmentioning

confidence: 99%

“…In the ‘100% variant’ (following the terminology of Tao ), we assume that

{∥ f ∥}_{U^{s} false[N false]} = 1

. It is then a simple exercise to show that

f (n) = e (p (n))

for all

n

, where

p \in R [x]

and

deg p < s

as above.…”

Section: Introductionmentioning

confidence: 99%

On uniformity of q‐multiplicative sequences

Fan

Konieczny

2019

Bull. London Math. Soc.

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We show that any q‐multiplicative sequence which is oscillating of order 1, that is, does not correlate with linear phase functions e2πinα (α∈R), is Gowers uniform of all orders, and hence in particular does not correlate with polynomial phase functions e2πipfalse(nfalse) (p∈R[x]). Quantitatively, we show that any q‐multiplicative sequence which is of Gelfond type of order 1 is automatically of Gelfond type of all orders. Consequently, any such q‐multiplicative sequence is a good weight for ergodic theorems. We also obtain combinatorial corollaries concerning linear patterns in sets which are described in terms of sums of digits.

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Higher Order Fourier Analysis

Cited by 97 publications

References 78 publications

Integrable communication channels and the nonlinear fourier transform

Integrable communication channels and the nonlinear fourier transform

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

On uniformity of q‐multiplicative sequences

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