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:

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

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

This extends naturally to higher dimensions. 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:

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

The criterion is named after, and was first formulated by, Hermann Weyl^[1] .

References

^ Weyl, H. (1916). "Ueber die Gleichverteilung von Zahlen mod. Eins,". Math. Ann. 77 (3): 313–352. doi:10.1007/BF01475864.