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:

:$lim\_\{n\; ightarrowinfty\}frac\{1\}\{n\}sum\_\{j=0\}^\{n-1\}e^\{2piimathell\; x\_\{j=0.$

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

:$x\_\{n\}inmathbb\{R\}^\{k\}$

is "equidistributed mod 1" if and only if we have:

:$lim\_\{n\; ightarrowinfty\}frac\{1\}\{n\}sum\_\{j=0\}^\{n-1\}e^\{2piimath(ell\_\{1\}x\_\{j\}^\{1\}+ell\_\{2\}x\_\{j\}^\{2\}+cdots+ell\_\{k\}x\_\{j\}^\{k\})\}=0.$

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

External links

* [http://mathworld.wolfram.com/WeylsCriterion.html Weyl's Criterion at Mathworld]
* [http://planetmath.org/encyclopedia/WeylsCriterion.html Weyl's Criterion at Planetmath]