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 $\ell$ we have:

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

Therefore distribution questions can be reduced to bounds on exponential sums, a fundamental and general method.

This extends naturally to higher dimensions. We say a sequence

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

is equidistributed mod 1 if and only if $\forall \ell\in\mathbb{Z}^{k}\backslash\{0\}$ we have:

$\lim_{n\rightarrow\infty}\frac{1}{n}\sum_{j=0}^{n-1}e^{2\pi\imath(\ell_{1}x_{j}^{1}+\ell_{2}x_{j}^{2}+\cdots+\ell_{k}x_{j}^{k})}=0.$

The criterion is named after, and was first formulated by, Hermann Weyl.

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