# 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

## Abstract

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 language English (US) Handbook of Discrete and Computational Geometry, Third Edition CRC Press 331-357 27 9781498711425 9781498711395 https://doi.org/10.1201/9781315119601 Published - Jan 1 2017

## All Science Journal Classification (ASJC) codes

• Computer Science(all)
• Mathematics(all)

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