Weyl’s criterion

Let {un} be a sequence of real numbers. Then {un} is uniformly distributed modulo 1 if and only if


for every nonzero integer k, where e(x)=exp(2πix).

Weyl’s criterion reduces the problem of uniform distributionMathworldPlanetmath of sequences to the problem of estimating certain exponential sums. Whereas the problem of estimating a family of exponential sums might seem harder at first, the exponential map has the multiplicative property which often makes the problem easier.

Example: If x is irrational, then the sequence {nx} is uniformly distributed modulo 1. Proof:


because the irrationality of x implies e(kx)1.


  • 1 Ð?. Ã?. Карацуба. ОÑ?новы аналитичеÑ?кой теории чиÑ?ел. Ð?аука, 1983. http://www.emis.de/cgi-bin/zmen/ZMATH/en/quick.html?type=html&an=0428.10019Zbl 0428.10019. For English translationMathworldPlanetmathPlanetmath see [2].
  • 2 A. A. Karatsuba. Basic analytic number theoryMathworldPlanetmath. Springer-Verlag, 1993. http://www.emis.de/cgi-bin/zmen/ZMATH/en/quick.html?type=html&an=0767.11001Zbl 0767.11001.
  • 3 Hugh L. Montgomery. Ten Lectures on the Interface Between Analytic Number Theory and Harmonic Analysis, volume 84 of Regional Conference Series in Mathematics. AMS, 1994. http://www.emis.de/cgi-bin/zmen/ZMATH/en/quick.html?type=html&an=0814.11001Zbl 0814.11001.
Title Weyl’s criterion
Canonical name WeylsCriterion
Date of creation 2013-03-22 14:17:31
Last modified on 2013-03-22 14:17:31
Owner bbukh (348)
Last modified by bbukh (348)
Numerical id 7
Author bbukh (348)
Entry type Theorem
Classification msc 11K06
Classification msc 11K38
Classification msc 11L03
Related topic UniformlyDistributed