474 XI Uniform Distribution and Ergodic Theory
and hence
cn=cD− (^1) n for everyn∈Zd.
Butcm=0 for some nonzerom∈Zd,sincefis not constant a.e., and|cn|→ 0
as|n|→∞,sincef∈L(X,B,λ).SincecD−km=cmfor every positive integerk,
it follows that the subscriptsD−kmare not all distinct. HenceDpm=mfor some
positive integerpandAhas an eigenvalue which is a root of unity.
(There are generalizations of Propositions 19 and 20 to translations and endomor-
phisms of any compact abelian groupX, with Haar measure in place of Lebesgue
measure.)
The preceding results havean application to the theory of ‘normal numbers’. In
fact, without any extra effort, we will consider also higher-dimensional generaliza-
tions. A vectorx∈Rdis said to benormal with respect to the matrix A,whereAis a
d×dmatrix of integers, if the sequence (Anx) is uniformly distributed mod 1.
Proposition 21Let A be a d×d matrix of integers. Then (λ-) almost all vectors
x∈Rdare normal with respect to A if and only if A is nonsingular and no eigenvalue
of A is a root of unity.
Proof IfAis nonsingular and no eigenvalue ofAis a root of unity then, by Proposi-
tion 20,RAis an ergodic measure-preserving transformation of the torusX=Rd/Zd.
Hence, by Proposition 18(ii), for each nonzerom∈Zd,
n−^1
n∑− 1
k= 0
e(m·Anx)→0asn→∞ for almost allx∈Rd.
SinceZdis countable, and the union of a countable number of sets of measure zero is
again a set of measure zero, it follows that, for almost allx∈Rd,
n−^1
n∑− 1
k= 0
e(m·Anx)→0asn→∞ for every nonzerom∈Zd.
Hence, by Theorem 2′, almost allx∈Rdare normal with respect toA.
IfAis singular then, by the remark following the proof of Proposition 9, nox∈Rd
is normal with respect toA. Suppose finally that some eigenvalue ofAis a root of unity.
Then there exists a positive integerpand a nonzero vectorz∈Zdsuch thatDpz=z,
whereD=At.If
f(x)=e(z·x)+e(z·Ax)+···+e(z·Ap−^1 x),
thenf(Ax)=f(x)and hence
n−^1
∑n−^1
k= 0
f(Akx)=f(x).
But ifxis normal with respect toAthen, by Theorem 1′,