476 XI Uniform Distribution and Ergodic Theory
Brown and Moran (1993) have shown, conversely, that ifA,Barecommuting d×d
nonsingular integer matrices, no eigenvalues of which are roots of unity, such that the
set of all vectors normal with respect toAcoincides with the set of all vectors normal
with respect toB,thenAp=Bqfor some positive integersp,q.
These results will now be specialized to the scalar case. A real numberxis said to
benormal to the base a,whereais an integer≥2, if the sequence (anx) is uniformly
distributed mod 1. It is readily shown thatxis normal to the baseaif and only if, in
the expansion ofxto the basea:
x= x +x 1 /a+x 2 /a^2 +···,wherexi∈{ 0 , 1 ,...,a− 1 }for alli≥1andxi=a−1 for at most finitely manyi,
every block of digits occurs with the proper frequency; i.e., for any positive integerk
and anya 1 ,...,ak∈{ 0 , 1 ,...,a− 1 }, the numberv(N)ofiwith 1≤i≤Nsuch that
xi=a 1 ,xi+ 1 =a 2 ,...,xi+k− 1 =ak,satisfiesv(N)/N→a−kasN→∞. By Proposition 21, almost all real numbersx
are normal to a given basea. The original proof of this by Borel (1909) was a forerun-
ner of Birkhoff’s ergodic theorem. (In fact Borel’s proof was faulty, but his paper was
influential. Borel used a different definition of normal number, but Wall (1949) showed
that it was equivalent to the definition in terms of uniform distribution adopted here.)
The first published proof of the scalar case of Corollary 23 was given by Schmidt
(1960), who also proved the scalar version of the result of Brown and Moran: the set
of all numbers normal to the baseacoincides with the set of all numbers normal to
the baseb,whereaandbare integers≥2, if and only ifap=bqfor some positive
integersp,q.
Although almost all real numbers are normal toevery basea, it is still not
known if such familiar irrational numbers as
√
2 ,eorπare normal to some base.
There are, however, various explicit constructions of normal numbers. In particular,
Champernowne (1933) showed that the real numberθwhose expansion to the base
10 is composed of the positive integers in their natural order, in other words,
θ= 0. 123456789101112 ..., is itself normal to the base 10.
(iii) LetAbe a set of finite cardinalityr, which for definiteness we take to be the set
{ 1 ,...,r},andletp 1 ,...,prbe positive real numbers with sum 1. IfB 0 is the family
of all subsets of the finite setAand if, for anyB 0 ∈B 0 , we putμ 0 (B 0 )=
∑
a∈B 0 pa,
thenμ 0 is a probability measure and (A,B 0 ,μ 0 ) is a probability space.
Now letXbe the set of all bi-infinite sequencesx=(...,x− 2 ,x− 1 ,x 0 ,x 1 ,x 2 ,...)
withxi∈Afor everyi∈Z. ThusXis the product of infinitely many copies ofA.We
construct aproduct measureonXin the following way.
For any finite sequence (a−m,...,a 0 ,...,am) withai ∈ Afor−m ≤i≤m,
define the (special)cylinder set[a−m,...,am]oforder mto be the set of allx∈X
such thatxi=aifor−m≤i≤m.Therearer^2 m+^1 distinct cylinder sets of orderm,
distinct cylinder sets are disjoint andXis the union of them all.
LetCmdenote the collection of all unions of distinct cylinder sets of orderm. Thus
X ∈Cmand, ifB∈Cm,thenBc =X\B ∈Cm. MoreoverB,C∈ Cmimplies
B∪C∈CmandB∩C∈Cm.IfB∈Cm,say