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- W2093506712 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 can be of the form [ n δ ] where δ is any positive real number and [ x ] denotes the integer part of x . More generally, the common difference can be of the form [ a ( n ) ] where a ( x ) is any function that is a member of a Hardy field and satisfies a ( x ) / x k → ∞ and a ( x ) / x k + 1 → 0 for some non-negative integer k . The proof combines a new structural result for Hardy sequences, techniques from ergodic theory, and some recent equidistribution results of sequences on nilmanifolds." @default.
- W2093506712 created "2016-06-24" @default.
- W2093506712 creator A5032701417 @default.
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- W2093506712 date "2009-09-01" @default.
- W2093506712 modified "2023-10-14" @default.
- W2093506712 title "A Hardy field extension of Szemerédi's theorem" @default.
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- W2093506712 doi "https://doi.org/10.1016/j.aim.2009.03.017" @default.
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