Number Theory: An Introduction to Mathematics

XI

Uniform Distribution and Ergodic Theory

A trajectory of a system which is evolving with time may be said to be ‘recurrent’ if it
keeps returning to any neighbourhood, however small, of its initial point, and ‘dense’
if it passes arbitrarily near to every point. It may be said to be ‘uniformly distributed’
if the proportion of time it spends in any region tends asymptotically to the ratio of the
volume of that region to the volume of thewhole space. In the present chapter these
notions will be made precise and some fundamental properties derived. The subject of
dynamical systems has its roots in mechanics, but we will be particularly concerned
with its applications in number theory.

1 UniformDistribution

Before introducing our subject, we establish the following interesting result:

Lemma 0Let J=[a,b]be a compact interval and fn:J→Ra sequence of non-
decreasing functions. If fn(t)→f(t)for every t∈Jasn→∞,where f:J→R
is a continuous function, then fn(t)→f(t)uniformlyon J.

Proof Evidently f is also nondecreasing. Furthermore, sinceJ is compact, f is
uniformly continuous onJ. It follows that, for anyε>0, there is a subdivision
a=t 0 <t 1 <···<tm=bsuch that

f(tk)−f(tk− 1 )<ε(k= 1 ,...,m).

We can choose a positive integerpso that, for alln>p,

|fn(tk)−f(tk)|<ε(k= 0 , 1 ,...,m).

Ift∈J,thent∈[tk− 1 ,tk]forsomek∈{ 1 ,...,m}. Hence

fn(t)−f(t)≤fn(tk)−f(tk)+f(tk)−f(tk− 1 )< 2 ε

and similarly

fn(t)−f(t)≥fn(tk− 1 )−f(tk− 1 )+f(tk− 1 )−f(tk)>− 2 ε.

Thus|fn(t)−f(t)|< 2 εfor everyt∈Jifn>p.

W.A. Coppel, Number Theory: An Introduction to Mathematics, Universitext,
DOI: 10.1007/978-0-387-89486-7_11, © Springer Science + Business Media, LLC 2009

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