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• W2950573819 abstract "In 1975 Szemer'edi 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 Szemer'edi's theorem can be of the form $[n^delta]$ where $delta$ 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 $a(x)/x^kto infty$ and $a(x)/x^{k+1}to 0$ 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.
• W2950573819 created "2019-06-27" @default.
• W2950573819 creator A5032701417 @default.
• W2950573819 creator A5042492763 @default.
• W2950573819 date "2008-02-20" @default.
• W2950573819 modified "2023-09-27" @default.
• W2950573819 title "A Hardy field extension of Szemeredi's Theorem" @default.
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• W2950573819 hasPublicationYear "2008" @default.
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