Equidistribution modulo one

Remarks

Example: for irrational $\alpha$, the sequence $x_n = n\alpha$ is equidistributed modulo one; this is the equidistribution theorem. Non-example: $x_n = n/2$ has $\{x_n\} \in \set{0, 1/2}$, so for $[a, b) = [0.1, 0.4)$ the count is $0$ for every $N$, and the limit $0$ differs from $0.3$. Weyl's criterion gives an equivalent form: the sequence is equidistributed modulo one if and only if $$\lim_{N \to \infty} \frac{1}{N} \sum_{n=1}^{N} e^{2\pi i h x_n} = 0$$ for every nonzero integer $h$. Another equivalent form: $\frac{1}{N} \sum_{n=1}^{N} f(\{x_n\}) \to \int_0^1 f(t)\,dt$ for every continuous function $f$ on $[0, 1]$ with $f(0) = f(1)$. Some authors also say uniformly distributed modulo one.