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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}rac{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}rac{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.

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External links

External links

★ Weyl's Criterion at Mathworld

★ Weyl's Criterion at Planetmath