Exact order of extreme [Formula: see text] discrepancy of infinite sequences in arbitrary dimension

We study the extreme [Formula: see text] discrepancy of infinite sequences in the d-dimensional unit cube, which uses arbitrary sub-intervals of the unit cube as test sets. This is in contrast to the classical star [Formula: see text] discrepancy, which uses exclusively intervals that are anchored i...

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Autores principales: Kritzinger, Ralph, Pillichshammer, Friedrich
Formato: Online Artículo Texto
Lenguaje:English
Publicado: Springer International Publishing 2022
Materias:
Article
Acceso en línea:https://www.ncbi.nlm.nih.gov/pmc/articles/PMC8881274/
https://www.ncbi.nlm.nih.gov/pubmed/35250035
http://dx.doi.org/10.1007/s00013-021-01688-9
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author Kritzinger, Ralph
Pillichshammer, Friedrich
author_facet Kritzinger, Ralph
Pillichshammer, Friedrich
author_sort Kritzinger, Ralph
collection PubMed
description We study the extreme [Formula: see text] discrepancy of infinite sequences in the d-dimensional unit cube, which uses arbitrary sub-intervals of the unit cube as test sets. This is in contrast to the classical star [Formula: see text] discrepancy, which uses exclusively intervals that are anchored in the origin as test sets. We show that for any dimension d and any [Formula: see text] , the extreme [Formula: see text] discrepancy of every infinite sequence in [Formula: see text] is at least of order of magnitude [Formula: see text] , where N is the number of considered initial terms of the sequence. For [Formula: see text] , this order of magnitude is best possible.
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spelling pubmed-88812742022-03-02 Exact order of extreme [Formula: see text] discrepancy of infinite sequences in arbitrary dimension Kritzinger, Ralph Pillichshammer, Friedrich Arch Math Article We study the extreme [Formula: see text] discrepancy of infinite sequences in the d-dimensional unit cube, which uses arbitrary sub-intervals of the unit cube as test sets. This is in contrast to the classical star [Formula: see text] discrepancy, which uses exclusively intervals that are anchored in the origin as test sets. We show that for any dimension d and any [Formula: see text] , the extreme [Formula: see text] discrepancy of every infinite sequence in [Formula: see text] is at least of order of magnitude [Formula: see text] , where N is the number of considered initial terms of the sequence. For [Formula: see text] , this order of magnitude is best possible. Springer International Publishing 2022-02-01 2022 /pmc/articles/PMC8881274/ /pubmed/35250035 http://dx.doi.org/10.1007/s00013-021-01688-9 Text en © The Author(s) 2022 https://creativecommons.org/licenses/by/4.0/Open AccessThis article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/ (https://creativecommons.org/licenses/by/4.0/) .
spellingShingle Article
Kritzinger, Ralph
Pillichshammer, Friedrich
Exact order of extreme [Formula: see text] discrepancy of infinite sequences in arbitrary dimension
title Exact order of extreme [Formula: see text] discrepancy of infinite sequences in arbitrary dimension
title_full Exact order of extreme [Formula: see text] discrepancy of infinite sequences in arbitrary dimension
title_fullStr Exact order of extreme [Formula: see text] discrepancy of infinite sequences in arbitrary dimension
title_full_unstemmed Exact order of extreme [Formula: see text] discrepancy of infinite sequences in arbitrary dimension
title_short Exact order of extreme [Formula: see text] discrepancy of infinite sequences in arbitrary dimension
title_sort exact order of extreme [formula: see text] discrepancy of infinite sequences in arbitrary dimension
topic Article
url https://www.ncbi.nlm.nih.gov/pmc/articles/PMC8881274/
https://www.ncbi.nlm.nih.gov/pubmed/35250035
http://dx.doi.org/10.1007/s00013-021-01688-9
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