Geometric discrepancy theory and uniform distribution

John Ralph Alexander, József Beck, William W.L. Chen

Research output: Chapter in Book/Report/Conference proceedingChapter

3 Scopus citations


A sequence s 1, s 2, … $ s_1, s_2,\ldots $ in U = [ 0, 1) $ { \text{ U}}=[0,1) $ is said to be uniformly distributed if, in the limit, the number of s j $ s_j $ falling in any given subinterval is proportional to its length. Equivalently, s 1, s 2, … $ s_1, s_2,\ldots $ is uniformly distributed if the sequence of equiweighted atomic probability measures µ N (s j) = 1 / N $ \mu _N(s_j)=1/N $, supported by the initial N-segments s 1, s 2, …, s N $ s_1, s_2,\ldots, s_N $, converges weakly to Lebesgue measure on U $ {{ \text{ U}}} $. This notion immediately generalizes to any topological space with a corresponding probability measure on the Borel sets.

Original languageEnglish (US)
Title of host publicationHandbook of Discrete and Computational Geometry, Third Edition
PublisherCRC Press
Number of pages27
ISBN (Electronic)9781498711425
ISBN (Print)9781498711395
StatePublished - Jan 1 2017

ASJC Scopus subject areas

  • Computer Science(all)
  • Mathematics(all)


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