480 XI Uniform Distribution and Ergodic Theory
Here it makes no difference if ‘integrable’ and ‘almost all’ refer to the invariant
measureμor to Lebesgue measure, since 1/ 2 ≤( 1 +x)−^1 ≤1.
Takingf to be the indicator function of the set{ξ ∈X:a 1 =m}, we see that
the asymptotic relative frequency of the positive integermamong the partial quotients
a 1 ,a 2 ,...is almost always
(log 2)−^1∫m− 1(m+ 1 )−^1( 1 +x)−^1 dx=(log 2)−^1 log((m+ 1 )^2 /(m(m+ 2 )).It follows, in particular, that almost allξ∈Xhave unbounded partial quotients.
Again, by takingf(ξ)=logξit may be shown that, for almost allξ∈X,
lim
n→∞
( 1 /n)logqn(ξ)=π^2 /(12 log 2),whereqn(ξ)is the denominator of then-th convergentpn/qnofξ. This was first proved
by L ́evy (1929).
In a letter to Laplace, Gauss (1812) stated that, for eachx∈( 0 , 1 ), the proportion
ofξ∈Xfor whichTnξ<xconverges asn→∞to log( 1 +x)/(log 2)and he asked
if Laplace could provide an estimate for the rapidity of convergence. If one writes
rn(x)=mn(x)−log( 1 +x)/(log 2),wheremn(x)is the Lebesgue measure of the set of allξ∈Xsuch thatTnξ<x,then
Gauss’s statement is thatrn(x)→0asn→∞and his question is, how fast?
Gauss’s statement was first proved by Kuz’min (1928), who also gave an estimate
for the rapidity of convergence. If one regards Gauss’s statement as a proposition in
ergodic theory, then one needs to know thatTis not only ergodic but evenmixing,i.e.
for allA,B∈B,
μ(T−nA∩B)→μ(A)μ(B) asn→∞.Kuz’min’s estimatern(x)=O(q√
n)for someq∈( 0 , 1 )was improved by L ́evy(1929) and Sz ̈usz (1961) torn(x)=O(qn)withq = 0 .7andq = 0 .485 respec-
tively. A substantial advance was made by Wirsing (1974). By means of an infinite-
dimensional generalization of a theorem of Perron (1907) and Frobenius (1908) on
positive matrices, he showed that
rn(x)=(−λ)nψ(x)+O(x( 1 −x)μn),whereψis a twice continuously differentiable function withψ( 0 ) =ψ( 1 )=0,
0 <μ<λandλ= 0. 303663 ....Wirsing’s analysis has been extended by Babenko
(1978) and Mayer (1990).
(v) Suppose we are given a system of ordinary differential equations
dx/dt=f(x), (†)wherex∈Rdandf:Rd→Rdis a continuously differentiable function. Then, for
anyx∈Rd, there is a unique solutionφt(x)of (†) such thatφ 0 (x)=x.