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Equidistribution


Prove that if (\xi_n) is a sequence of real numbers such that the fractional parts of \xi_{n+h}-\xi_n are equidistributed in [0,1] for every positive integer h, then the fractional parts of \xi_n are also equidistributed in [0,1].

Use this to prove that if P is a non-constant polynomial with real coefficients, with at least one of these coefficients irrational, then the fractional parts of P(n) are equidistributed in [0,1].

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  1. Dan Schwarz
    April 30, 2011 at 10:25 pm

    I have a good idea the solution iis Van der Corput’s difference theorem; see this Terence Tao write-up http://terrytao.wordpress.com/2008/06/14/the-van-der-corputs-trick-and-equidistribution-on-nilmanifolds/

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