Relevant bibliographies by topics / Green-Tao theorem

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  2. Dissertations / Theses
  3. Book chapters

Academic literature on the topic 'Green-Tao theorem'

Author: Grafiati
Published: 4 June 2021
Last updated: 1 February 2022

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Journal articles on the topic "Green-Tao theorem"

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Conlon, David, Jacob Fox, and Yufei Zhao. "The Green-Tao theorem: an exposition." EMS Surveys in Mathematical Sciences 1, no. 2 (2014): 257–91. http://dx.doi.org/10.4171/emss/6.

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Lê, Thái Hoàng. "Green–Tao theorem in function fields." Acta Arithmetica 147, no. 2 (2011): 129–52. http://dx.doi.org/10.4064/aa147-2-3.

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ZHAO, YUFEI. "An arithmetic transference proof of a relative Szemerédi theorem." Mathematical Proceedings of the Cambridge Philosophical Society 156, no. 2 (November 14, 2013): 255–61. http://dx.doi.org/10.1017/s0305004113000662.

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AbstractRecently, Conlon, Fox and the author gave a new proof of a relative Szemerédi theorem, which was the main novel ingredient in the proof of the celebrated Green–Tao theorem that the primes contain arbitrarily long arithmetic progressions. Roughly speaking, a relative Szemerédi theorem says that if S is a set of integers satisfying certain conditions, and A is a subset of S with positive relative density, then A contains long arithmetic progressions, and our recent results show that S only needs to satisfy a so-called linear forms condition.This paper contains an alternative proof of the new relative Szemerédi theorem, where we directly transfer Szemerédi's theorem, instead of going through the hypergraph removal lemma. This approach provides a somewhat more direct route to establishing the result, and it gives better quantitative bounds.The proof has three main ingredients: (1) a transference principle/dense model theorem of Green–Tao and Tao–Ziegler (with simplified proofs given later by Gowers, and independently, Reingold–Trevisan–Tulsiani–Vadhan) applied with a discrepancy/cut-type norm (instead of a Gowers uniformity norm as it was applied in earlier works); (2) a counting lemma established by Conlon, Fox and the author; and (3) Szemerédi's theorem as a black box.
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Rimanić, Luka, and Julia Wolf. "Szemerédi's Theorem in the Primes." Proceedings of the Edinburgh Mathematical Society 62, no. 2 (November 19, 2018): 443–57. http://dx.doi.org/10.1017/s0013091518000561.

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AbstractGreen and Tao famously proved in 2005 that any subset of the primes of fixed positive density contains arbitrarily long arithmetic progressions. Green had previously shown that, in fact, any subset of the primes of relative density tending to zero sufficiently slowly contains a three-term progression. This was followed by work of Helfgott and de Roton, and Naslund, who improved the bounds on the relative density in the case of three-term progressions. The aim of this note is to present an analogous result for longer progressions by combining a quantified version of the relative Szemerédi theorem given by Conlon, Fox and Zhao with Henriot's estimates of the enveloping sieve weights.
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SANDERS, TOM. "Approximate groups and doubling metrics." Mathematical Proceedings of the Cambridge Philosophical Society 152, no. 3 (December 13, 2011): 385–404. http://dx.doi.org/10.1017/s0305004111000740.

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AbstractWe develop a version of Freĭman's theorem for a class of non-abelian groups, which includes finite nilpotent, supersolvable and solvable A-groups. To do this we have to replace the small doubling hypothesis with a stronger relative polynomial growth hypothesis akin to that in Gromov's theorem (although with an effective range), and the structures we find are balls in (left and right) translation invariant pseudo-metrics with certain well behaved growth estimates.Our work complements three other recent approaches to developing non-abelian versions of Freĭman's theorem by Breuillard and Green, Fisher, Katz and Peng, and Tao.
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Kra, Bryna. "The Green-Tao Theorem on arithmetic progressions in the primes: an ergodic point of view." Bulletin of the American Mathematical Society 43, no. 01 (October 6, 2005): 3–24. http://dx.doi.org/10.1090/s0273-0979-05-01086-4.

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Chen, Yong-Gao, and Ying Shi. "Dynamics of the $w$ function and the Green-Tao theorem on arithmetic progressions in the primes." Proceedings of the American Mathematical Society 136, no. 07 (March 4, 2008): 2351–57. http://dx.doi.org/10.1090/s0002-9939-08-09207-1.

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Pambuccian, Victor. "The Green-Tao theorem on primes in arithmetical progressions in the positive cone of $\mathbb Z [X]$." Elemente der Mathematik 69, no. 1 (2014): 30–32. http://dx.doi.org/10.4171/em/242.

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Sun, Yu-Chen, and Hao Pan. "The Green–Tao theorem for primes of the form $$x^2+y^2+1$$ x 2 + y 2 + 1." Monatshefte für Mathematik 189, no. 4 (December 15, 2018): 715–33. http://dx.doi.org/10.1007/s00605-018-1245-0.

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EVEN-ZOHAR, CHAIM. "On Sums of Generating Sets in ℤ2n." Combinatorics, Probability and Computing 21, no. 6 (August 3, 2012): 916–41. http://dx.doi.org/10.1017/s0963548312000351.

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Let A and B be two affinely generating sets of ℤ2n. As usual, we denote their Minkowski sum by A+B. How small can A+B be, given the cardinalities of A and B? We give a tight answer to this question. Our bound is attained when both A and B are unions of cosets of a certain subgroup of ℤ2n. These cosets are arranged as Hamming balls, the smaller of which has radius 1.By similar methods, we re-prove the Freiman–Ruzsa theorem in ℤ2n, with an optimal upper bound. Denote by F(K) the maximal spanning constant |〈 A 〉|/|A| over all subsets A ⊆ ℤ2n with doubling constant |A+A|/|A| ≤ K. We explicitly calculate F(K), and in particular show that 4K/4K ≤ F(K)⋅(1+o(1)) ≤ 4K/2K. This improves the estimate F(K) = poly(K)4K, found recently by Green and Tao [17] and by Konyagin [23].
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Dissertations / Theses on the topic "Green-Tao theorem"

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Henriot, Kevin. "Structures linéaires dans les ensembles à faible densité." Thèse, Paris 7, 2014. http://hdl.handle.net/1866/11116.

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Réalisé en cotutelle avec l'Université Paris-Diderot.
Nous présentons trois résultats en combinatoire additive, un domaine récent à la croisée de la combinatoire, l'analyse harmonique et la théorie analytique des nombres. Le thème unificateur de notre thèse est la détection de structures additives dans les ensembles arithmétiques à faible densité, avec un intérêt particulier pour les aspects quantitatifs. Notre première contribution est une estimation de densité améliorée pour le problème, initié entre autres par Bourgain, de trouver une longue progression arithmétique dans un ensemble somme triple. Notre deuxième résultat consiste en une généralisation des bornes de Sanders pour le théorème de Roth, du cas d'un ensemble dense dans les entiers à celui d'un ensemble à faible croissance additive dans un groupe abélien arbitraire. Finalement, nous étendons les meilleures bornes quantitatives connues pour le théorème de Roth dans les premiers, à tous les systèmes d'équations linéaires invariants par translation et de complexité un.
We present three results in additive combinatorics, a recent field at the interface of combinatorics, harmonic analysis and analytic number theory. The unifying theme in our thesis is the detection of additive structure in arithmetic sets of low density, with an emphasis on quantitative aspects. Our first contribution is an improved density estimate for the problem, initiated by Bourgain and others, of finding a long arithmetic progression in a triple sumset. Our second result is a generalization of Sanders' bounds for Roth's theorem from the dense setting, to the setting of small doubling in an arbitrary abelian group. Finally, we extend the best known quantitative results for Roth's theorem in the primes, to all translation-invariant systems of equations of complexity one.
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Book chapters on the topic "Green-Tao theorem"

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Kullmann, Oliver. "Green-Tao Numbers and SAT." In Theory and Applications of Satisfiability Testing – SAT 2010, 352–62. Berlin, Heidelberg: Springer Berlin Heidelberg, 2010. http://dx.doi.org/10.1007/978-3-642-14186-7_32.

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