In this section we will study more closely the statistical properties of irrational rotation dynamics. Fix an irrational number $\alpha$ and define $T$ on $[0,1)$ by \[ T(x) = x + \alpha \bmod 1 \] for all $x \in [0,1)$. Given $E \subset [0,1)$ we investigate for each $N \in \N$ the statistical quantity \[ \dfrac{|\{ 0 \le n \le N-1 : T^n(x) \in E \}|}{N} \] and in particular wish to understand whether its limit exists as $N \to \infty$.
How can we analyse this average? There is a fruitful way to rewrite it in terms of functions. We have \[ \dfrac{|\{ 0 \le n \le N-1 : T^n(x) \in E \}|}{N} = \dfrac{1}{N} \sum_{n=0}^{N-1} 1_E(T^n(x)) \] for all $N \in \N$ and this suggests considering more generally averages of the form \[ \dfrac{1}{N} \sum_{n=0}^{N-1} f(T^n(x)) \] for $f : [0,1) \to \R$.
First we note that the limit of the above quantity, if it exists, is well-behaved under uniform convergence. Recall that $f_k$ converges to $g$ uniformly, if, for every $\epsilon > 0$ there is $K \in \N$ such that \[ k \ge K \Rightarrow \| f_k - g\|_\mathsf{u} \l \epsilon \] holds. Here the expression \[ \| h \|_\mathsf{u} = \sup \{ |h(x)| : x \in X \} \] is the uniform norm of $h : X \to \C$. It is convenient to write uniform convergence in terms of the uniform norm, but we are not describing a new type of convergence; we have only rephrased the definition of uniform convergence we are used to from real analysis. In particular, for example, it is true that if $f_k \to g$ uniformly and each $f_k$ is continuous then $g$ will also be continuous.
Fix a sequence $k \mapsto f_k$ of functions from $X$ to $\C$ that converges uniformly to a function $g : X \to \C$. If \[ \lim_{N \to \infty} \dfrac{1}{N} \sum_{n=0}^{N-1} f_k(T^n x) \] exists for every $k \in \N$ then \[ \lim_{N \to \infty} \dfrac{1}{N} \sum_{n=0}^{N-1} g(T^n x) = \lim_{k \to \infty} \lim_{N \to \infty} \dfrac{1}{N} \sum_{n=0}^{N-1} f_k(T^n x) \] and in particular the limit on the left-hand side exists.
Since $k \mapsto f_k$ is in particular a Cauchy sequence for the uniform norm one can show \[ k \mapsto \alpha(k) = \lim_{N \to \infty} \dfrac{1}{N} \sum_{n=0}^{N-1} f_k(T^n x) \] is a Cauchy sequence of real numbers and therefore converges to a limit, say $\beta$.
Fix $\epsilon > 0$. Choose $k$ so large that $\| f_k - g \|_\mathsf{u} \l \tfrac{\epsilon}{2}$ and \[ \left| \beta - \lim_{N \to \infty} \dfrac{1}{N} \sum_{n=0}^{N-1} f_k(T^n (x)) \right| \l \frac{\epsilon}{2} \] both hold. Choose then $M$ so large that \[ N \ge M \Rightarrow \left| \alpha - \dfrac{1}{N} \sum_{n=0}^{N-1} f_k(T^n(x)) \right| \l \frac{\epsilon}{2} \] holds. For every $N \ge M$ we have \[ \left| \alpha - \dfrac{1}{N} \sum_{n=0}^{N-1} g(T^n(x)) \right| \l \epsilon \] by the triangle inequality. ▮
On $[0,1)$ there is a special collection of functions for which the average is straightforward to analyze and which approximate all other continuous functions with respect to the uniform norm. These are the exponential functions \[ \psi_k(x) = \exp(2 \pi i k x) \] defined for each $k \in \Z$.
For every continuous function $g : X \to \C$ and every $\epsilon \g 0$ there are $r(1),\dots,r(s)$ in $\Z$ and $c(1),\dots,c(s)$ in $\C$ such that \[ \| c(1) \psi_{r(1)} + \cdots + c(s) \psi_{r(s)} - g \|_\mathsf{u} \l \epsilon \] holds.
We will take this result for granted. It can be deduced from the Stone-Weierstrass theorem or proved directly via analysis of the Fejér kernel. ▮
Let us then analyze the average \[ \dfrac{1}{N} \sum_{n=0}^{N-1} \psi_k(T^n(x)) \] as $N \to \infty$ for each $k \in \Z$. Since $\psi_0 = 1$ we clearly have \[ \lim_{N \to \infty} \dfrac{1}{N} \sum_{n=0}^{N-1} \psi_0(T^n(x)) = 1 \] for all $x \in X$ and proceed assuming $k \ne 0$. We can calculate \[ \begin{aligned} & \sum_{n=0}^{N-1} \psi_k(T^n(x)) \\ = {} & \sum_{n=0}^{N-1} \exp(2 \pi i k (x + n \alpha)) \\ = {} & \exp(2 \pi i k x) \dfrac{1 - \exp(2 \pi i k N \alpha)}{1 - \exp(2 \pi i k \alpha)} \end{aligned} \] using the geometric series formula because $\alpha$ is irrational. Now \[ \left| \sum_{n=0}^{N-1} \psi_k(T^n(x)) \right| \le \dfrac{2}{|1 - \exp(2 \pi i k \alpha)|} \] for all $N \in \N$ and, since the upper-bound is independent of $N$ we have \[ \lim_{N \to \infty} \dfrac{1}{N} \sum_{n=0}^{N-1} \psi_k(T^n(x)) = 0 \] for all non-zero $k \in \Z$ and all $x \in X$.
Combining all of the above, we conclude that \[ \lim_{N \to \infty} \dfrac{1}{N} \sum_{n=0}^{N-1} f(T^n(x)) \] exists for every $x \in X$ and every continuous function $f : X \to \C$. In fact, more is true. We have \[ \lim_{N \to \infty} \dfrac{1}{N} \sum_{n=0}^{N-1} f(T^n(x)) = \int\limits_0^1 f(x) \intd x = \int f \intd \mu \] for every continuous function $f : X \to \C$ and every $x \in X$.
To finish, let us return to $f = 1_E$ where $E = [a,b)$ is a sub-interval. We cannot approximate $1_{[a,b)}$ by continuous functions with respect to the uniform norm because the function $1_{[a,b)}$ is not continuous! However, we can find for every $\epsilon \ge 0$ continuous functions $g,h$ from $X$ to $[0,1]$ with \[ g \le 1_{[a,b)} \le h \] and \[ 0 \le \int h-g \intd \mu \l \epsilon \] from which \[ \dfrac{1}{N} \sum_{n=0}^{N-1} g(T^n(x)) \le \dfrac{1}{N} \sum_{n=0}^{N-1} 1_{[a,b)}(T^n(x)) \le \dfrac{1}{N} \sum_{n=0}^{N-1} h(T^n(x)) \] follows for all $N \in \N$ and all $x \in [0,1)$. Taking the limit as $N \to \infty$ the expression \[ \lim_{N \to \infty} \dfrac{1}{N} \sum_{n=0}^{N-1} 1_{[a,b)}(T^n(x)) \] is forced to exist as it is sandwiched between \[ \int g \intd \mu \] and \[ \int h \intd \mu \] which, as we make $\epsilon$ smaller and smaller, both converge to $b-a$. Thus we have proved the following theorem.
Fix $\alpha$ irrational. For all $0 \le a \l b \le 1$ we have \[ \lim_{N \to \infty} \dfrac{|\{0 \le n \le N-1 : a \le n \alpha \bmod 1 \l b\}|}{N} = b-a \] for all $x \in [0,1)$.
We say that the sequence $n \alpha$ is uniformly distributed mod 1. It is remarkable that the result is true simultaneously for all intervals $[a,b) \subset [0,1)$.