Number Theory: An Introduction to Mathematics

458 XI Uniform Distribution and Ergodic Theory

By the mean value theorem, the hypotheses of Proposition 10 are certainly satisfied
ifξn=f(n),wherefis a differentiable function such thatf′(t)→0 monotonically
ast→∞andt|f′(t)|→∞ast→∞. Consequently the sequence(anα) is uni-
formly distributed mod 1 ifa=0and0<α<1, and the sequence(a(logn)α)is
uniformly distributed mod 1 ifa=0andα>1. By using van der Corput’s difference
theorem and an inductive argument starting from Proposition 10, it may be further
shown that the sequence(anα)is uniformly distributed mod 1 for anya=0andany
α>0 which is not an integer.
It has been shown by Kemperman (1973) that ‘if’ may be replaced by ‘if and only
if’ in the statement of Proposition 10. Consequently the sequence(a(logn)α)is not
uniformly distributed mod 1 if 0<α≤1.
The theory of uniform distribution has an application, and its origin, in astronomy.
In his investigations on the secular perturbations of planetary orbits Lagrange (1782)
was led to the problem ofmean motion:if

z(t)=

∑n

k= 1

ρke(ωkt+αk),

whereρk>0andαk,ωk ∈R(k = 1 ,...,n), doest−^1 argz(t)have a finite limit
ast→+∞? It is assumed thatz(t)never vanishes and argz(t)is then defined by
continuity. (Zeros ofz(t)can be admitted by writingz(t)=ρ(t)e(φ(t)),whereρ(t)
andφ(t)are continuous real-valued functions andρ(t)is required to change sign at a
zero ofz(t)of odd multiplicity.)
In the astronomical application argz(t)measures the longitude of the perihelion of
the planetary orbit. Lagrange showed that the limit

μ= lim

t→+∞

t−^1 argz(t)

does exist whenn=2 and also, for arbitraryn, when someρkexceeds the sum of all
the others. The only planets which do not satisfy this second condition are Venus and
Earth. Lagrange went on to say that, when neither of the two conditions was satisfied,
the problem was “very difficult and perhaps impossible”.
There was no further progress until the work of Bohl (1909), who tookn= 3
and considered the non-Lagrangian case when there exists a triangle with sidelengths
ρ 1 ,ρ 2 ,ρ 3. He showed that the limitμexists ifω 1 ,ω 2 ,ω 3 are linearly independent
over the rational fieldQand thenμ=λ 1 ω 1 +λ 2 ω 2 +λ 3 ω 3 ,whereπλ 1 ,πλ 2 ,πλ 3
are the angles of the triangle with sidelengthsρ 1 ,ρ 2 ,ρ 3. In the course of the proof he
stated and proved Proposition 3 (without formulating the general concept of uniform
distribution).
Using his earlier results on uniform distribution, Weyl (1938) showed that the limit
μexists ifω 1 ,...,ωnare linearly independent over the rational fieldQand then

μ=λ 1 ω 1 +···+λnωn,

whereλk≥ 0 (k= 1 ,...,n)and

∑n
k= 1 λk=1. The coefficientsλkdepend only on
theρ’s, not on theα’s orω’s, and there is even an explicit expression forλk, involving
Bessel functions, which is derived from the theory of random walks.
Finally, it was shown by Jessen and Tornehave (1945) that the limitμexists for
arbitraryωk∈R.