2009
DOI: 10.1016/j.aim.2009.03.017
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A Hardy field extension of Szemerédi's theorem

Abstract: In 1975 Szemerédi proved that a set of integers of positive upper density contains arbitrarily long arithmetic progressions. Bergelson and Leibman showed in 1996 that the common difference of the arithmetic progression can be a square, a cube, or more generally of the form p(n) where p(n) is any integer polynomial with zero constant term. We produce a variety of new results of this type related to sequences that are not polynomial. We show that the common difference of the progression in Szemerédi's theorem ca… Show more

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Cited by 34 publications
(44 citation statements)
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“…Remark 1.7. For additional multiple recurrence results dealing with non-polynomial functions from Hardy fields, see [13,14,15,12].…”
Section: Introductionmentioning
confidence: 99%
“…Remark 1.7. For additional multiple recurrence results dealing with non-polynomial functions from Hardy fields, see [13,14,15,12].…”
Section: Introductionmentioning
confidence: 99%
“…The motivation for this work comes from another line of thought. Since PS(α) is the (rounded) image of N under the Hardy field function 1 x → x α , it is known to be a so-called set of multiple recurrence in ergodic theory (see [8]); thus, for example, every E ⊆ N with lim sup N →∞ E ∩ {1, . .…”
Section: Introductionmentioning
confidence: 99%
“…, N } N > 0 contains arbitrarily long arithmetic progressions with step size in PS(α). That PS(α) is a set of multiple recurrence follows from it containing "many divisible polynomial patterns" (see [8], Section 5); in particular, when 1 < α < 2, the set PS(α) contains arbitrarily long arithmetic progressions and arithmetic progressions of every sufficiently large step.…”
Section: Introductionmentioning
confidence: 99%
“…Remark: For a Szemerédi type result involving multiples of a tempered Hardy function see the recent preprint [12].…”
Section: Examplementioning
confidence: 99%