# Weyl's criterion

In mathematics, in the theory of Diophantine approximation, Weyl's criterion states that a sequence $(x_{n})$ of real numbers is equidistributed mod 1 if and only if for all non-zero integers $\ell$

$\lim_{n\rightarrow\infty}\frac{1}{n}\sum_{j=0}^{n-1}e^{2\pi i \ell x_{j}}=0.$

The criterion is named after, and was first formulated by, Hermann Weyl.[1] It allows to equidistribution questions to bounds on exponential sums, a fundamental and general method.

The criterion extends naturally to higher dimensions. A sequence

$x_{n}\in\mathbb{R}^{k}$

is equidistributed mod 1 if and only if for any $\ell\in\mathbb{Z}^{k}$ which is not the zero vector,

$\lim_{n\rightarrow\infty}\frac{1}{n}\sum_{j=0}^{n-1}e^{2\pi i (\ell \cdot x_{j})}=0.$