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