Weyl's criterion

Remarks

Equidistribution modulo one is equivalent to $\frac{1}{N} \sum_{n \le N} f(x_n) \to \int_0^1 f(t) \, dt$ for every continuous function $f \colon \R \to \C$ of period $1$. The functions $t \mapsto e^{2\pi i h t}$ are such functions, with integral $0$ for $h \ne 0$, which gives one direction. For the other direction, the condition gives the limit for every trigonometric polynomial. Uniform approximation by the Weierstrass theorem extends it to every continuous periodic $f$. Finally, squeezing the indicator of an interval between continuous functions gives the counting condition. Example: for irrational $\alpha$ and $x_n = n \alpha$, the sum is a geometric sum bounded by $2 / \abs{1 - e^{2\pi i h \alpha}}$, so the averages tend to $0$; this proves the equidistribution theorem. For a rational $\alpha = p/q$ the test fails at $h = q$, where every term equals $1$. The same criterion holds in $\R^k$, with $h$ running over nonzero vectors of $\Z^k$ and $h x_n$ replaced by the dot product $h \cdot x_n$.

Going deeper

For a student

The criterion turns a question about counting points in intervals into estimates of exponential sums. It gives a short proof that $(n\alpha)$ is equidistributed for irrational $\alpha$. With Weyl differencing, it also shows that $p(n)$ is equidistributed for a real polynomial $p$ with an irrational coefficient other than the constant term. Its proof is a first example of approximating indicator functions by trigonometric polynomials, a basic method of Fourier analysis.