Erdős and Arithmetic Progressions

Gowers, Timothy

doi:10.1007/978-3-642-39286-3_8

Cited by 12 publications

(14 citation statements)

References 36 publications

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“…Here I include many results showing these new developments (and leave out certain parts covered by other surveys of this volume, see Katona e.g., [214]. I also cut short describing areas that are covered by the very recent survey papers of the Erdős Centennial volume, e.g., Gowers [189], Rödl and Schacht [303] or Füredi and myself [180], . .…”

Section: Prefacementioning

confidence: 99%

“…the corresponding chapter of the book of Graham, Rothschild and Spencer [190]. At the same time, there are fascinating approaches to this field using deep analysis, due to Gowers, and others, 27 see recent papers of Gowers [186], or an even newer paper of Gowers [189] on these types of problems, on arithmetic progressions.…”

Section: Remarks 72 (A) Szemerédi's Theorem Is One Of the Roots Of mentioning

confidence: 99%

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Paul Turán’s influence in combinatorics

Simonovits¹

2013

Number Theory, Analysis, and Combinatorics

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This paper is a survey on the topic in extremal graph theory influenced directly or indirectly by Paul Turán. While trying to cover a fairly wide area, I will try to avoid most of the technical details. Areas covered by detailed fairly recent surveys will also be treated only briefly. The last part of the survey deals with random ±1 matrices, connected to some early results of Szekeres and Turán.

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Section: Prefacementioning

confidence: 99%

Section: Remarks 72 (A) Szemerédi's Theorem Is One Of the Roots Of mentioning

confidence: 99%

Paul Turán’s influence in combinatorics

Simonovits¹

2013

Number Theory, Analysis, and Combinatorics

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“…For random ±1 sequences of length l the discrepancy grows as l 1/2+o (1) and the explicit constructions of a sequence with slowly growing discrepancy at the rate of log 3 l have been demonstrated [14,8]. It is known [17] that discrepancy of any infinite ±1 sequence can not be bounded by 1, that is, Erdős's conjecture holds for the particular case C = 1.…”

Section: Introductionmentioning

confidence: 99%

A SAT Attack on the Erdős Discrepancy Conjecture

Konev

Lisitsa

2014

Lecture Notes in Computer Science

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In 1930s Paul Erdős conjectured that for any positive integer C in any infinite ±1 sequence (xn) there exists a subsequence x d , x 2d , x 3d , . . . , x kd , for some positive integers k and d, such that | k i=1 x id |> C. The conjecture has been referred to as one of the major open problems in combinatorial number theory and discrepancy theory. For the particular case of C = 1 a human proof of the conjecture exists; for C = 2 a bespoke computer program had generated sequences of length 1124 of discrepancy 2, but the status of the conjecture remained open even for such a small bound. We show that by encoding the problem into Boolean satisfiability and applying the state of the art SAT solver, one can obtain a discrepancy 2 sequence of length 1160 and a proof of the Erdős discrepancy conjecture for C = 2, claiming that no discrepancy 2 sequence of length 1161, or more, exists. We also present our partial results for the case of C = 3.

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“…This was in fact the preferred form of the Fourier-analytic reduction obtained by the Polymath5 project [19], [10]. It is conceivable that some refinement of the analysis in this paper in fact yields a bound of the form (2.4), though this seems to require removing the logarithmic averaging from Theorem 1.10, as well as avoiding the use of Lemma 4.1 below.…”

Section: Fourier Analytic Reductionmentioning

confidence: 99%

The Erdős discrepancy problem

Tao¹

2016

Discrete Analysis

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We show that for any sequence f (1), f (2), . . . taking values in {−1, +1}, the discrepancyof f is infinite. This answers a question of Erdős. In fact the argument also applies to sequences f taking values in the unit sphere of a real or complex Hilbert space.The argument uses three ingredients. The first is a Fourier-analytic reduction, obtained as part of the Polymath5 project on this problem, which reduces the problem to the case when f is replaced by a (stochastic) completely multiplicative function g. The second is a logarithmically averaged version of the Elliott conjecture, established recently by the author, which effectively reduces to the case when g usually pretends to be a modulated Dirichlet character. The final ingredient is (an extension of) a further argument obtained by the Polymath5 project which shows unbounded discrepancy in this case.

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Erdős and Arithmetic Progressions

Abstract: Two of Erdős's most famous conjectures concern arithmetic progressions. In this paper we discuss some of the progress that has been made on them.

Cited by 12 publications

References 36 publications

Paul Turán’s influence in combinatorics

Paul Turán’s influence in combinatorics

A SAT Attack on the Erdős Discrepancy Conjecture

The Erdős discrepancy problem

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