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An important result of Weyl states that for every sequence(   a  n   )    n  ⩾  1of distinct positive integers the sequence of fractional parts of(   a  n   α  )    n  ⩾  1is uniformly distributed modulo one for almost all   α  . However, in general it is a very hard problem to calculate the precise order of convergence of the discrepancy of(   {   a  n   α  }   )    n  ⩾  1for almost all   α  . In particular, it is very difficult to give sharp lower bounds for the speed of convergence. Until now this was only carried out for lacunary sequences(   a  n   )    n  ⩾  1and for some special cases such as the Kronecker sequence(   {  n  α  }   )    n  ⩾  1or the sequence(   {   n  2   α  }   )    n  ⩾  1    . In the present paper we answer the question for a large class of sequences(   a  n   )    n  ⩾  1including as a special case all polynomialsa  n   =  P   (  n  )with    P  ∈  Z  [  x  ]    of degree at least 2.

mathematika

METRIC RESULTS ON THE DISCREPANCY OF SEQUENCES     (   a  n   α  )    n  ⩾  1     MODULO ONE FOR INTEGER SEQUENCES     (   a  n   )    n  ⩾  1     OF POLYNOMIAL GROWTH