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.