uniformly distributed

Let {un} be a sequence of real numbers. For 0α<β1 put


The sequence {un} is uniformly distributed modulo 1 if


for all 0α<β1. In other words a sequence is uniformly distributed modulo 1 if each subinterval of [0,1] gets its “fair share” of fractional parts of {un}.

More generally, a sequence {un} of points in a finite measure space (X,𝒜,μ) is uniformly distributed with respect to a family of sets 𝒜 if

limNcard{n[1..N]:unS}N=μ(S)μ(X)  for every S.


  • 1 William Chen. Lectures on irregularities of point distribution. Available at http://www.maths.mq.edu.au/ wchen/ln.htmlhttp://www.maths.mq.edu.au/ wchen/ln.html, 2000.
  • 2 Hugh L. Montgomery. Ten Lectures on the Interface Between Analytic Number TheoryMathworldPlanetmath 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 uniformly distributed
Canonical name UniformlyDistributed
Date of creation 2013-03-22 14:17:29
Last modified on 2013-03-22 14:17:29
Owner bbukh (348)
Last modified by bbukh (348)
Numerical id 6
Author bbukh (348)
Entry type Definition
Classification msc 11K06
Classification msc 11K38
Synonym equidistributed
Related topic WeylsCriterion