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
} 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
(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 .
The following assertions concerning the sequence {u n}
are equivalent:
(a) The sequence {u
} 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 ) =
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.
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