Number Theory: An Introduction to Mathematics

1 Uniform Distribution 451

If f is a continuous function of period 1 then, by the Weierstrass approximation
theorem, for anyε>0 there exists a trigonometric polynomialg(t)such that
|f(t)−g(t)|<εfor everyt∈I. Hence

∣

∣

∣

∣N

− 1

∑N

n= 1

f({ξn})−

∫

I

f(t)dt

∣

∣

∣

∣

≤

∣

∣

∣

∣N

− 1

∑N

n= 1

(f({ξn})−g({ξn}))

∣

∣

∣

∣+

∣

∣

∣

∣N

− 1

∑N

n= 1

g({ξn})−

∫

I

g(t)dt

∣

∣

∣

∣

+

∣

∣

∣

∣

∫

I

(g(t)−f(t))dt

∣

∣

∣

∣

< 2 ε+

∣

∣

∣

∣N

− 1

∑N

n= 1

g({ξn})−

∫

I

g(t)dt

∣

∣

∣

∣

< 3 ε for all largeN.

Thus (1) holds for every continuous functionfof period 1.
Finally, ifχα,βis the function defined in the proof of Theorem 1 then, for any
ε>0, there exist continuous functionsf 1 ,f 2 of period 1 such that

f 1 (t)≤χα,β(t)≤f 2 (t) for everyt∈I

and
∫

I

(f 2 (t)−f 1 (t))dt<ε,

from which it follows similarly that

N−^1

∑N

n= 1

χα,β({ξn})→

∫

I

χα,β(t)dt asN→∞.

Thus the sequence(ξn)is uniformly distributed mod 1.

Weyl’s criterion, as Theorem 2 is usually called, immediately implies Bohl’s result:

Proposition 3Ifξis irrational, the sequence(nξ)is uniformly distributed mod 1.

Proof For any nonzero integerh,

e(hξ)+e( 2 hξ)+···+e(Nhξ)=(e((N+ 1 )hξ)−e(hξ))/(e(hξ)− 1 ).

Hence

∣

∣

∣

∣N

− 1

∑N

n= 1

e(hnξ)

∣

∣

∣

∣≤^2 |e(hξ)−^1 |

− (^1) N− (^1) ,
and the result follows from Theorem 2.