The primes contain arbitrarily long polynomial progressions

Tao, Terence; Ziegler, Tamar

doi:10.1007/s11511-008-0032-5

Cited by 97 publications

(144 citation statements)

References 35 publications

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“…, a+ P k (m)} ⊆ A. More recently [23] the same result was established where A is only assumed to have positive relative density in the set of prime numbers. Other polynomializations of versions of Szemerédi's Theorem can be found in [9] and [10].…”

Section: Theorem 12 (Finite Unions Theorem)mentioning

confidence: 64%

Polynomial extensions of the Milliken-Taylor Theorem

Bergelson

Hindman

Williams

2014

Trans. Amer. Math. Soc.

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Abstract. Milliken-Taylor systems are some of the most general infinitary configurations that are known to be partition regular. These are sets of the. . , Fm are increasing finite nonempty subsets of N}, where a 1 , a 2 , . . . , am ∈ Z with am > 0 and xn ∞ n=1 is a sequence in N. That is, if p(y 1 , y 2 , . . . , ym) = m i=1 a i y i is a given linear polynomial and a finite coloring of N is given, one gets a sequence xn ∞ n=1 such that all sums of the form p( t∈F 1 xt, . . . , t∈Fm xt) are monochromatic. In this paper we extend these systems to images of very general extended polynomials. We work with the Stone-Čech compactification βF of the discrete space F of finite subsets of N, whose points we take to be the ultrafilters on F . We utilize a simply stated result about the tensor products of ultrafilters and the algebraic structure of βF .

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Section: Theorem 12 (Finite Unions Theorem)mentioning

confidence: 64%

Polynomial extensions of the Milliken-Taylor Theorem

Bergelson

Hindman

Williams

2014

Trans. Amer. Math. Soc.

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“…The Green-Tao theorem has inspired a variety of other results in ergodic theory and number theory, such as Frantzikinakis-Host-Kra [61] or Tao-Ziegler [188]. The focus of this note will remain on connections to harmonic analysis, and thus we return to restriction theory for the last time.…”

Section: The Sum-product Problemmentioning

confidence: 99%

From harmonic analysis to arithmetic combinatorics

Łaba

2007

Bull. Amer. Math. Soc.

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Arithmetic combinatorics, or additive combinatorics, is a fast developing area of research combining elements of number theory, combinatorics, harmonic analysis and ergodic theory. Its arguably best-known result, and the one that brought it to global prominence, is the proof by Ben Green and Terence Tao of the long-standing conjecture that primes contain arbitrarily long arithmetic progressions. There are many accounts and expositions of the Green-Tao theorem, including the articles by Kra [119] and Tao [182] in the Bulletin.The purpose of the present article is to survey a broader, highly interconnected network of questions and results, built over the decades and spanning several areas of mathematics, of which the Green-Tao theorem is a famous descendant. An old geometric problem lies at the heart of key conjectures in harmonic analysis. A major result in partial differential equations invokes combinatorial theorems on intersecting lines and circles. An unexpected argument points harmonic analysts towards additive number theory, with consequences that could have hardly been anticipated.We will not try to give a comprehensive survey of harmonic analysis, combinatorics, or additive number theory. We will not even be able to do full justice to our specific areas of focus, instead referring the reader to the more complete expositions and surveys listed in Section 7. Our goal here is to emphasize the connections between these areas; we will thus concentrate on relatively few problems, chosen as much for their importance to their fields as for their links to each other. The article is written from the point of view of an analyst who, in the course of her work, was gradually introduced to the questions discussed here and found them fascinating. We hope that the reader will enjoy a taste of this experience.

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“…Tao and Ziegler [325] (see also [252]), via a transference principle for polynomial configurations, extended the Green-Tao theorem to cover polynomial progressions: Let A ⊂ P be a set of primes of positive relative upper density in the primes, i.e., lim sup N →∞ |A ∩…”

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 primes contain arbitrarily long polynomial progressions

Cited by 97 publications

References 35 publications

Polynomial extensions of the Milliken-Taylor Theorem

Polynomial extensions of the Milliken-Taylor Theorem

From harmonic analysis to arithmetic combinatorics

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

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