Weyl’s criterion

Let $\{u_n\}$ be a sequence of real numbers. Then $\{u_n\}$ isuniformly distributed modulo $1$ if and only if

$$\underset{N\to \mathrm{\infty }}{lim}\frac{1}{N}\sum _{n=1}^{N}e(k{u}_n)=0$$

for every nonzero integer $k$, where $e(x)=\exp (2\pi ix)$.

Weyl’s criterion reduces the problem of uniform distribution ofsequences to the problem of estimating certain exponential sums.Whereas the problem of estimating a family of exponential sumsmight seem harder at first, the exponential map has themultiplicative property which often makes the problem easier.

Example: If $x$ is irrational, then the sequence $\{nx\}$is uniformly distributed modulo $1$. Proof:

$$\left|\sum _{n=1}^{N}e(knx)\right|=\left|\frac{e(k(N+1)x)-e(kx)}{e(kx)-1}\right|\le \frac{2}{\left|e(kx)-1\right|}=O_k(1)$$

because the irrationality of $x$ implies $e(kx)\ne 1$.