464 XI Uniform Distribution and Ergodic Theory
Thus ifC∗is the least upper bound for all admissible values ofCin Schmidt’s
result then, by what has been said, 0. 060 ...≤C∗≤ 0. 223 .... It is natural to ask:
what is the exact value ofC∗, and is there a sequence(ξn)for which it is attained?
The notion of discrepancy is easily extended to higher dimensions by defining
the discrepancy of a finite set of vectorsx 1 ,...,xNin thed-dimensional unit cube
Id=I×···×Ito be
D∗N(x 1 ,...,xN)= sup
0 <αk≤ 1 (k= 1 ,...,d)|φa(N)/N−α 1 ···αd|,wherexn=(ξn(^1 ),...,ξn(d)),a=(α 1 ,...,αd)andφa(N)is the number of positive
integersn≤Nsuch that 0≤ξn(k)<αkfor everyk∈{ 1 ,...,d}.
Ford>1 there is no simple reformulation of the definition analogous to Proposi-
tion 11, but many results do carry over. In particular, Proposition 15 was generalized
and applied to the numerical evaluation of multiple integrals by Hlawka (1961/62).
Indeed this application has greater value in higher dimensions, where other methods
perform poorly.
For the application one requires a set of vectorsx 1 ,...,xN∈Idwhose discrep-
ancyD∗N(x 1 ,...,xN)is small. A simple procedure for obtaining such a set, which is
most useful when the integrand is smooth and has period 1 in each of its variables, is
the method of ‘good lattice points’ introduced by Korobov (1959). Here, for a suitably
choseng∈Zd, one takesxn={(n− 1 )g/N}(n= 1 ,...,N). A result of Niederreiter
(1986) implies that, for everyd≥2andeveryN≥2, one can choosegso that
ND∗N≤( 1 +logN)d+d 2 d.The van der Corput sequence has also been generalized to any finite number of
dimensions by Halton (1960). He defined an infinite sequence (xn) of vectors inRdfor
which
lim
N→∞NδN/(logN)d<∞.It is conjectured that for eachd>1(asford=1) there exists an absolute constant
Cd>0 such that
lim
N→∞NδN/(logN)d≥Cdfor every infinite sequence(xn)of vectors inRd. However, the best known result
remains that of Roth (1954), in which the exponentdis replaced byd/2.
3 Birkhoff’s Ergodic Theorem
In statistical mechanics there is a procedure for calculating the physical properties of
a system by simply averaging over all possible states of the system. To justify this
procedure Boltzmann (1871) introduced what he later called the ‘ergodic hypothesis’.
In the formulation of Maxwell (1879) this says that “the system, if left to itself in its