We will now apply the machinery already developed to a few concrete problems.
Proposition 1 Let
be a compact abelian group and
the rotation by
. Then
is uniquely ergodic (with the Haar measure invariant) if
is dense in
.
The proof is straightforward. Suppose is invariant with respect to rotations by
. Then for
, we have
and hence
for any , which means that
must be Haar measure (which is unique).
Corollary 2 An irrational rotation of the unit circle
is uniquely ergodic.
Application: Equidistribution
Theorem 3 Let
be irrational and let
be continuous and
-periodic. Then
The proof of this theorem is evident when one considers as a function
, notes that an irrational rotation is uniquely ergodic, and applies the criterion from last time.
Say that a sequence is equidistributed if for every interval
,
Theorem 4 (Weyl equidistribution theorem) If
is irrational, then the sequence of fractional parts
is equidistributed. More generally, for any
,
is equidistributed.
This would follow immediately if we had an analog of the boxed limit condition above for a step function, which we could take as the characteristic function of any interval
, extended by periodicity. However, the previous theorem holds only for
continuous. This is no matter; if
is a step function, there are continuous
with
and we find that for a step function ,
and a similar inequality for the , which proves the boxed equality for step functions, and implies the theorem.
Application: First digits of powers of 2
Now for another application.
I claim that ergodic theory will enable us to find the asyptotic proportions of first digits in the powers of 2. As it happens, they obey Benford’s law.
Now the integer has first digit
if and only if there exists
with
i.e.
Theorem 5 Let
be an integer. Then the asymptotic probability that
has first digit
is
.
Indeed, , and the fractional part of this is in
with probability
by Weyl’s theorem.
Of course this generalizes to other bases, exponents, etc.
April 3, 2010 at 9:56 pm
Hi! Akhil Mathew : Mathematics do not have national boundaries I am a chinese ungraduate students of ShanDong Normal University I appreciate you for your advanced study and deep thought I wil be very glad of being good friends with you in mathematics Emil : WaiCheung2010@yahoo.com MSN : waicheung2010@hotmail.com I am steping into Algebraic Geometry and the application in Number Theory And I am prepared to get into the graduate test of Peking University Best Wishes