Number Theory: An Introduction to Mathematics

454 XI Uniform Distribution and Ergodic Theory

Proof By takingζn=e(ξn)in Lemma 4 we obtain, for 1≤M≤N,

N−^2

∣

∣

∣

∣

∑N

n= 1

e(ξn)

∣

∣

∣

∣

2

≤ 2 (M+N− 1 )M−^2 N−^2

M∑− 1

m= 1

(M−m)

∣

∣

∣

∣

N∑−m

n= 1

e(ξn+m−ξn)

∣

∣

∣

∣

+(M+N− 1 )M−^1 N−^1.

KeepingMfixed and lettingN→∞,weget

lim

N→∞

N−^2

∣

∣

∣

∣

∑N

n= 1

e(ξn)

∣

∣

∣

∣

2

≤M−^1.

ButMcan be chosen as large as we please.

An immediate consequence is van der Corput’sdifference theorem:

Proposition 6The real sequence(ξn)is uniformly distributed mod 1 if, for each pos-
itive integer m, the sequence(ξn+m−ξn)is uniformly distributed mod 1.

Proof If the sequences(ξn+m −ξn)are uniformly distributed mod 1 then, by
Theorem 2,

N−^1

∑N

n= 1

e(h(ξn+m−ξn))→0asN→∞

for all integersh= 0 ,m>0. Replacingξnbyhξnin Corollary 5 we obtain, for all
integersh=0,

N−^1

∑N

n= 1

e(hξn)→0asN→∞.

Hence, by Theorem 2 again, the sequence(ξn)is uniformly distributed mod 1.

The sequence(nξ), withξirrational, shows that we cannot replace ‘if’ by ‘if and
only if’ in the statement of Proposition 6. Weyl’s result will now be derived from
Proposition 6:

Proposition 7If

f(t)=αrtr+αr− 1 tr−^1 +···+α 0

is any polynomial with real coefficientsαksuch thatαkis irrational for at least one
k> 0 , then the sequence(f(n))is uniformly distributed mod 1.

Proof Ifr=1, then the result holds by the same argument as in Proposition 3. We
assume thatr>1,αr=0 and the result holds for polynomials of degree less thanr.
For any positive integerm,

gm(t)=f(t+m)−f(t)