Geometric Discrepancy Theory and Uniform Distribution

Authored by: J. Ralph Alexander , József Beck , William W.L. Chen

Discrete and Computational Geometry

Print publication date:  April  2004
Online publication date:  April  2004

Print ISBN: 9781584883012
eBook ISBN: 9781420035315
Adobe ISBN:

10.1201/9781420035315.ch13

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Abstract

A sequence s 1, s 2, … in U = [0, 1) is said to be uniformly distributed if, in the limit, the number of sj falling in any given subinterval is proportional to its length. Equivalently, s 1, s 2, … is uniformly distributed if the sequence of equiweighted atomic probability measures ?N (sj ) = 1/N, supported by the initial N-segments s 1, s 2, …, s N, converges weakly to Lebesgue measure on U. This notion immediately generalizes to any topological space with a corresponding probability measure on the Borel sets.

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