Weyl's Criterion
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High Quality Content by WIKIPEDIA articles! In mathematics, in the theory of diophantine approximation, Weyl's criterion states that a sequence (xn) of real numbers is equidistributed mod 1 if and only if for all non-zero integers ell we have: lim_{nrightarrowinfty}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 forall ellinmathbb{Z}^{k}backslash{0} we have: ...
High Quality Content by WIKIPEDIA articles! In mathematics, in the theory of diophantine approximation, Weyl's criterion states that a sequence (xn) of real numbers is equidistributed mod 1 if and only if for all non-zero integers ell we have: lim_{nrightarrowinfty}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 forall ellinmathbb{Z}^{k}backslash{0} we have: lim_{nrightarrowinfty}frac{1}{n}sum_{j=0}^{n-1}e^{2piimath(ell cdot x_{j})}=0. The criterion is named after, and was first formulated by, Hermann Weyl.
