Chapter 1 . Unifor m Distributio n 1. Qualitativ e theory . W e le t T denot e th e circl e group , T = R/Z . Sup - pose tha t w e are give n a sequence {u n } o f points o f T. Fo r 0 a 1 put Z(N a) = car d {n e Z : I n N,0 un a (mo d I)} . We say tha t th e sequenc e {u n } i s uniformly distributed i f (1) li m ±Z(N a) = a N—oo IS for every a, 0 a 1 . Le t UN denote the measure that place s unit point-masse s at th e point s u\,U2, ... ,UN Th e Fourie r transfor m o f thi s measur e i s define d to b e UN(k)= f e{-ka)dU N N (2) =VJe(-A u n ). n = l Here e(^) = e 2nie denote s the complex exponential with period 1 . Wey l [13 , 14 ] characterized uniforml y distribute d sequence s a s follows . WEYL'S CRITERION . The following assertions concerning the sequence {u n } are equivalent: (a) The sequence {u n } is uniformly distributed (b) For each integer k ^ 0 , UN{k) = o(N) as N oo (c) / / F is properly Riemann-integrable on T then 1 N r (3) K m T X K ) = F(a)da. By a simpl e compactnes s argumen t i t ma y b e see n tha t i f {u n } i s uniforml y distributed the n th e limi t (1 ) i s attained uniforml y i n a. Fo r 0 a 1 we pu t D(N a) = Z(N a)-Na. l http://dx.doi.org/10.1090/cbms/084/01
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