Number Theory: An Introduction to Mathematics

450 XI Uniform Distribution and Ergodic Theory

A converse of Theorem 1 has been proved by de Bruijn and Post (1968): if a func-
tionf:I→Chas the property that

lim

N→∞

N−^1

∑N

n= 1

f({ξn})

exists for every sequence(ξn)which is uniformly distributed mod 1, thenfis Riemann
integrable.
In the statement of the next result, and throughout the rest of the chapter, we use
the abbreviation

e(t)=e^2 πit.

In the proof of the next result we use theWeierstrass approximation theorem: any con-
tinuous functionf :I→Cof period 1 is the uniform limit of a sequence(fn)of
trigonometric polynomials. In fact, as Fej ́er (1904) showed, one can takefnto be the
arithmetic mean(S 0 +···+Sn− 1 )/n,where

Sm=Sm(x):=

∑m

h=−m

che(hx)

is them-th partial sum of the Fourier series forf. This yields the explicit formula

fn(x)=

∫

I

Kn(x−t)f(t)dt,

where

Kn(u)=(sin^2 nπu)/(nsin^2 πu).

Theorem 2A real sequence(ξn)is uniformly distributed mod 1 if and only if, for
every integer h= 0 ,

N−^1

∑N

n= 1

e(hξn)→ 0 as N→∞. (2)

Proof If the sequence(ξn)is uniformly distributed mod 1 then, by taking f(t)=
e(ht)in Theorem 1 we obtain (2) since, for every integerh=0,
∫

I

e(ht)dt= 0.

Conversely, suppose (2) holds for every nonzero integerh. Then, by linearity, for
any trigonometric polynomial

g(t)=

∑m

h=−m

bhe(ht)

we have

N−^1

∑N

n= 1

g({ξn})→b 0 =

∫

I

g(t)dt asN→∞.