We will now apply the machinery already developed to a few concrete problems.

Proposition 1Let 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 2An irrational rotation of the unit circle is uniquely ergodic.

**Application: Equidistribution **

Theorem 3Let 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 5Let 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