Equidistribution theorem

Remarks

A short proof uses Weyl's criterion. For a nonzero integer $h$, the number $e^{2\pi i h \alpha}$ is not $1$, since $h\alpha \notin \Z$. The sum $\sum_{n=1}^{N} e^{2\pi i h n \alpha}$ is a geometric sum, so its absolute value is at most $2 / \abs{1 - e^{2\pi i h \alpha}}$, a bound independent of $N$. Dividing by $N$ gives a limit $0$, which is Weyl's condition. The hypothesis that $\alpha$ is irrational is necessary: if $\alpha = p/q$ in lowest terms, then $n\alpha$ takes only $q$ values modulo one. Example: $\log_{10} 2$ is irrational, so the fractional parts of $n \log_{10} 2$ are equidistributed. Hence the leading digit of $2^n$ is $k$ for a proportion $\log_{10}(1 + 1/k)$ of the exponents $n$; for instance the digit $1$ occurs about $30.1\%$ of the time.

Going deeper

For a student

The theorem is the simplest case of Weyl's results on polynomial sequences. If a real polynomial $p$ has at least one irrational coefficient other than the constant term, then $(p(n))$ is equidistributed modulo one. Its proof shows the general pattern: equidistribution follows from cancellation in exponential sums. It strengthens Kronecker's theorem that $(n\alpha)$ is dense modulo one, and it expresses the unique ergodicity of the irrational rotation of the circle.