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
for any , which means that must be Haar measure (which is unique).
Corollary 2 An irrational rotation of the unit circle is uniquely ergodic.
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.