Number Theory: An Introduction to Mathematics

4 Applications 475

n−^1

∑n−^1

k= 0

f(Akx)→

∫

X

fdλ= 0.

Sincefis not zero a.e., it follows that the set of allxwhich are normal with respect
toAdoes not have full measure.

We consider next when normality with respect to one matrix coincides with
normality with respect to another matrix.

Proposition 22Let A be a d×d nonsingular matrix of integers, no eigenvalue of
which is a root of unity. Then, for any positive integer q , the vector x∈Rdis normal
with respect to Aqif and only if it is normal with respect to A.

Proof It follows at once from Proposition 9 that ifxis normal with respect toA,then
it is also normal with respect toAq.
Suppose, on the other hand, thatxis normal with respect toAq. Then, by Theorem
2 ′, for every nonzero vectorm∈Zd,

N−^1

N∑− 1

n= 0

e(m·Anqx)→0asN→∞.

PutD=At.SinceDis a nonsingular matrix of integers,Djmis a nonzero vector in
Zdfor any integerj≥0 and hence

N−^1

N∑− 1

n= 0

e(m·Anq+jx)=N−^1

N∑− 1

n= 0

e(Djm·Anqx)→0asN→∞.

Adding these relations forj= 0 , 1 ,...,q−1 and dividing byq, we obtain

(Nq)−^1

Nq∑− 1

n= 0

e(m·Anx)→0asN→∞.

Since the sum of at mostqtermse(m·Anx)has absolute value at mostqit follows
that, also without restrictingNto be a multiple ofq,

N−^1

N∑− 1

n= 0

e(m·Anx)→0asN→∞.

Hence, by Theorem 2′,xis normal with respect toA.

Corollary 23Let A be a d×d nonsingular integer matrix, no eigenvalue of which is a
root of unity, and let B be a d×d integer matrix such that Ap=Bqfor some positive
integers p,q. Then x∈Rdis normal with respect to A if and only if x is normal with
respect to B.

Proof This follows at once from Proposition 22, since the hypotheses imply that also
Bis nonsingular and has no eigenvalue which is a root of unity.