A general rearrangement theorem for sequences
type: Item
instance of: scholarly article
title: A general rearrangement theorem for sequences
zbMATH Open document ID: 0536.54020
DOI: 10.1007/BF01190956
author: Harald Niederreiter
published in: Archiv der Mathematik
publication date: 1984-01-01T00:00:00Z
review text: Let X be a compact metric space with metric d, and let \(\omega_ 1=(x_ n)\) and \(\omega_ 2=(y_ n)\) be sequences of elements of X. Then it is shown that the following two conditions are equivalent: (i) there exists a permutation \(\tau\) of the set of positive integers such that \(\lim_{n\to \infty}d(x_ n,y_{\tau(n)})=0\); (ii) \(\omega_ 1\) and \(\omega_ 2\) have the same accumulation points. From this result all the standard rearrangement theorems in the theory of uniform distribution of sequences can be obtained as simple consequences. For instance, it follows easily that if \(\mu\) is a Borel probability measure in X, then a sequence \(\omega\) of elements of X has a \(\mu\)-uniformly distributed (\(\mu\)-well distributed, completely \(\mu\)-uniformly distributed) rearrangement if and only if every point of the support of \(\mu\) is an accumulation point of \(\omega\).
Mathematics Subject Classification ID: 54E45
zbMATH DE Number: 3852000
zbMATH Keywords: rearrangement
MaRDI profile type: Publication
cites work: Q5550510
cites work: Q5621409
cites work: Folgen auf kompakten Räumen
cites work: Folgen auf kompakten Räumen. II
cites work: Q4767350
cites work: Q4063531
full work available at URL: https://doi.org/10.1007/bf01190956
OpenAlex ID: W2055980404
description: scientific article; zbMATH DE number 3852000