Number Theory: An Introduction to Mathematics

1 Uniform Distribution 449

Theorem 1A real sequence(ξn)is uniformly distributed mod 1 if and only if, for
every function f:I→Cwhich is Riemann integrable,

N−^1

∑N

n= 1

f({ξn})→

∫

I

f(t)dt as N→∞. (1)

Proof For anyα,β∈ Iwithα<β,letχα,βdenote theindicator functionof the
interval [α,β),i.e.

χα,β(t)=1forα≤t<β,

=0otherwise.

Since
∫

I

χα,β(t)dt=β−α,

the definition of uniform distribution can be rephrased by saying that the sequence(ξn)
is uniformly distributed mod 1 if and only if, for all choices ofαandβ,

N−^1

∑N

n= 1

χα,β({ξn})→

∫

I

χα,β(t)dt asN→∞.

Thus the sequence(ξn)is certainly uniformly distributed mod 1 if (1) holds for every
Riemann integrable functionf.
Suppose now that the sequence(ξn)is uniformly distributed mod 1. Then (1) holds
not only for every functionf =χα,β, but also for every finite linear combination
of such functions, i.e. for everystep-function f. But, for any real-valued Riemann
integrable functionfand anyε>0, there exist step-functionsf 1 ,f 2 such that

f 1 (t)≤f(t)≤f 2 (t) for everyt∈I

and
∫

I

(f 2 (t)−f 1 (t))dt<ε.

Hence

N−^1

∑N

n= 1

f({ξn})−

∫

I

f(t)dt≤N−^1

∑N

n= 1

f 2 ({ξn})−

∫

I

f 2 (t)dt+ε

< 2 ε for all largeN,

and similarly

N−^1

∑N

n= 1

f({ξn})−

∫

I

f(t)dt>− 2 ε for all largeN.

Thus (1) holds when the Riemann integrable functionf is real-valued and also, by
linearity, when it is complex-valued.