Number Theory: An Introduction to Mathematics

456 XI Uniform Distribution and Ergodic Theory

Since

q−^1

∑q

k= 1

e(nk/q)=1ifn≡0modq,

=0ifn≡0modq,

we can write

S=(qN)−^1

∑qN

n= 1

e(mξn+r)

∑q

k= 1

e(nk/q)

=(qN)−^1

∑q

k= 1

∑qN

n= 1

e(mη(nk)),

wherewehaveput

ηn(k)=ξn+r+nk/mq.

By hypothesis, for every positive integerh, the sequence

ηn(k+)h−ηn(k)=ξn+h+r−ξn+r−hk/mq

is uniformly distributed mod 1. Henceη(nk)is uniformly distributed mod 1, by Propo-
sition 6. Thus, for eachk∈{ 1 ,...,q},

(qN)−^1

∑qN

n= 1

e(mη(nk))→0asN→∞,

and consequently alsoS→0asN→∞.

As an application of Proposition 8 we prove

Proposition 9Let A be a d×d matrix of integers, no eigenvalue of which is a root
of unity. If, for some x∈Rd, the sequence(Anx)is uniformly distributed mod 1 then,
for any integers q> 0 and r≥ 0 , the sequence(Aqn+rx)is also uniformly distributed
mod 1.

Proof It follows from Theorem 2′that, for any nonzero vectorm∈Zd, the scalar
sequenceξn=m·Anxis uniformly distributed mod 1. For any positive integerh,the
sequence

ξn+h−ξn=m·(Ah−I)Anx=(Ah−I)tm·Anx

has the same form as the sequenceξn, since the hypotheses ensure that(Ah−I)tmis
a nonzero vector inZd. Hence the sequenceξn+h−ξnis uniformly distributed mod 1.
Therefore, by Proposition 8, the sequenceξqn+r=m·Aqn+rxis uniformly distributed
mod 1, and thus the sequenceAqn+rxis uniformly distributed mod 1.