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

Proposition 1 Let {G} be a compact abelian group and {T} the rotation by {a \in G}. Then {T} is uniquely ergodic (with the Haar measure invariant) if {a^{\mathbb{Z}}} is dense in {G}.

 

The proof is straightforward. Suppose {\mu} is invariant with respect to rotations by {a}. Then for {f \in C(G)}, we have

\displaystyle \int f(a^m x ) d \mu = \int f(x) d \mu, \quad \forall m \in \mathbb{Z}

and hence

\displaystyle \int f(bx ) d \mu = \int f(x) d \mu, \quad \forall m \in \mathbb{Z},

for any {b \in G}, which means that {\mu} must be Haar measure (which is unique).

Corollary 2 An irrational rotation of the unit circle {S^1} is uniquely ergodic.

 

Application: Equidistribution

 

Theorem 3 Let {\xi \in \mathbb{R}} be irrational and let {f: \mathbb{R} \rightarrow \mathbb{C}} be continuous and {2 \pi }-periodic. Then\displaystyle \boxed{ \lim_{N \rightarrow \infty} \frac{1}{N} \sum_{i=0}^{N-1} f( n \xi) = \int_0^1 f(x) dx .}

 

The proof of this theorem is evident when one considers {f} as a function {S^1 \rightarrow \mathbb{C}}, notes that an irrational rotation is uniquely ergodic, and applies the criterion from last time.

Say that a sequence {s_n \in [0,1]} is equidistributed if for every interval {I \subset [0,1]},

\displaystyle \lim_{N \rightarrow \infty} \frac{1}{N} \mathrm{card} \{ n: 0 \leq n \leq N-1, s_n \in I \} = |I|.

Theorem 4 (Weyl equidistribution theorem) If {\xi} is irrational, then the sequence of fractional parts {\{n \xi\}} is equidistributed. More generally, for any {x \in \mathbb{R}}, {\{ x + n \xi \}} is equidistributed.

 

This would follow immediately if we had an analog of the boxed limit condition above for {f} a step function, which we could take as the characteristic function of any interval {I}, extended by periodicity. However, the previous theorem holds only for {f} continuous. This is no matter; if {f} is a step function, there are continuous {f_1, f_2} with

\displaystyle f_1 \leq f \leq f_2, \quad \int ( f_2 - f_1 ) dx < \epsilon

and we find that for a step function {f},

\displaystyle \limsup_N \frac{1}{N} \sum_{i=0}^{N-1} f( n \xi) \leq \int_0^1 f(x) dx + \epsilon

and a similar inequality for the {\liminf}, 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 {x} has first digit {k \in \{0,1, \dots, 9\}} if and only if there exists {m} with { k 10^m \leq x < (k+1) 10^m,} i.e.

\displaystyle \{ \log_{10} x \} \in \left[ \log_{10} k, \log_{10}(k+1) \right).

Theorem 5 Let {n} be an integer. Then the asymptotic probability that {2^m n, m \in \mathbb{N}} has first digit {k} is {\log_{10}(1+1/k)}.

 

Indeed, {\log_{10}(2^mn) = m \log_{10} 2 + \log_{10}n}, and the fractional part of this is in {\left[ \log_{10} k, \log_{10}(k+1) \right)} with probability {\log_10(1+1/k)} by Weyl’s theorem.

Of course this generalizes to other bases, exponents, etc.

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