Integer Sum Sets Containing Long Arithmetic Progressions

Freiman, Gregory A.; Halberstam, H.; Ruzsa, Imre Z.

doi:10.1112/jlms/s2-46.2.193

Cited by 38 publications

(33 citation statements)

References 7 publications

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“…In the abelian setting the non-local version of this result is in the same vein as a result of Freiman, Halberstam and Ruzsa [17] that finds long arithmetic progressions or Bohr sets in A + A + A (see also [44,Theorem 4.43]); the best bounds currently known in this direction are due to Sanders [36].…”

Section: Proposition 13 (Lsupporting

confidence: 57%

A Probabilistic Technique for Finding Almost-Periods of Convolutions

2010

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Abstract. We introduce a new probabilistic technique for finding 'almost-periods' of convolutions of subsets of groups. This gives results similar to the Bogolyubov-type estimates established by Fourier analysis on abelian groups but without the need for a nice Fourier transform to exist. We also present applications, some of which are new even in the abelian setting. These include a probabilistic proof of Roth's theorem on three-term arithmetic progressions and a proof of a variant of the Bourgain-Green theorem on the existence of long arithmetic progressions in sumsets A + B that works with sparser subsets of {1, . . . , N } than previously possible. In the non-abelian setting we exhibit analogues of the Bogolyubov-Freiman-Halberstam-Ruzsa-type results of additive combinatorics, showing that product sets A 1 · A 2 · A 3 and A 2 · A −2 are rather structured, in the sense that they contain very large iterated product sets. This is particularly so when the sets in question satisfy small-doubling conditions or high multiplicative energy conditions. We also present results on structures in A · B.Our results are 'local' in nature, meaning that it is not necessary for the sets under consideration to be dense in the ambient group. In particular, our results apply to finite subsets of infinite groups provided they 'interact nicely' with some other set.

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Section: Proposition 13 (Lsupporting

confidence: 57%

A Probabilistic Technique for Finding Almost-Periods of Convolutions

2010

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“…Let us, for instance, address the problem of estimating the length of the longest arithmetic progression in lA (where A is a subset of [n]), as a function of l, n and |A|. In special cases sharp results have been obtained, thanks to the works of several researchers, including Bourgain, Freiman, Halberstam, Ruzsa and Sárközy [2], [6], [8], [17]. Our method, combined with additional arguments, allows us to derive a sharp bound for this length for a wide range of l and |A|.…”

Section: Introductionmentioning

confidence: 99%

Finite and infinite arithmetic progressions in sumsets

Szemerédi

2006

Ann. Math.

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“…также более ранние работы [7], [8] и недавнюю работу [9]). Сформулируем один из основных результатов статьи [6].…”

Section: и д шкредовunclassified

О Множествах Больших Тригонометрических Сумм

Шкредов¹,

Shkredov²

2008

Izv. RAN. Ser. Mat.

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О множествах больших тригонометрических суммДоказано существование нетривиальных решений уравнения r1 + r2 = r3 + r4, где r1, r2, r3, r4 принадлежат множеству R больших коэффициен-тов Фурье некоторого подмножества A из Z/N Z. Из этого утверждения следует, что множество R имеет сильные аддитивные свойства. Обсужда-ются обобщения и приложения полученных результатов.Библиография: 26 наименований.

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Integer Sum Sets Containing Long Arithmetic Progressions

Cited by 38 publications

References 7 publications

A Probabilistic Technique for Finding Almost-Periods of Convolutions

A Probabilistic Technique for Finding Almost-Periods of Convolutions

Finite and infinite arithmetic progressions in sumsets

О Множествах Больших Тригонометрических Сумм

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