M. Levin’s construction of absolutely normal numbers with very low discrepancy

Among the currently known constructions of absolutely normal numbers, the one given by Mordechay Levin in 1979 achieves the lowest discrepancy bound. In this work we analyze this construction in terms of computability and computational complexity. We show that, under basic assumptions, it yields a c...

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Detalles Bibliográficos
Autores: Alvarez, Nicolás Alejandro, Becher, Veronica Andrea
Tipo de recurso: artículo
Estado:Versión publicada
Fecha de publicación:2017
País:Argentina
Institución:Consejo Nacional de Investigaciones Científicas y Técnicas
Repositorio:CONICET Digital (CONICET)
Idioma:inglés
OAI Identifier:oai:ri.conicet.gov.ar:11336/42845
Acceso en línea:http://hdl.handle.net/11336/42845
Access Level:acceso abierto
Palabra clave:NORMAL NUMBERS
DISCREPANCY
ALGORITHMS
https://purl.org/becyt/ford/1.2
https://purl.org/becyt/ford/1
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spelling M. Levin’s construction of absolutely normal numbers with very low discrepancyAlvarez, Nicolás AlejandroBecher, Veronica AndreaNORMAL NUMBERSDISCREPANCYALGORITHMShttps://purl.org/becyt/ford/1.2https://purl.org/becyt/ford/1Among the currently known constructions of absolutely normal numbers, the one given by Mordechay Levin in 1979 achieves the lowest discrepancy bound. In this work we analyze this construction in terms of computability and computational complexity. We show that, under basic assumptions, it yields a computable real number. The construction does not give the digits of the fractional expansion explicitly, but it gives a sequence of increasing approximations whose limit is the announced absolutely normal number. The nth approximation has an error less than 2–2n. To obtain the $ nth approximation the construction requires, in the worst case, a number of mathematical operations that is doubly exponential in n. We consider variants on the construction that reduce the computational complexity at the expense of an increment in discrepancy.Fil: Alvarez, Nicolás Alejandro. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Bahía Blanca. Instituto de Ciencias e Ingeniería de la Computación. Universidad Nacional del Sur. Departamento de Ciencias e Ingeniería de la Computación. Instituto de Ciencias e Ingeniería de la Computación; ArgentinaFil: Becher, Veronica Andrea. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Departamento de Computación; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; ArgentinaAmerican Mathematical Society2017-03info:eu-repo/semantics/articleinfo:eu-repo/semantics/publishedVersionhttp://purl.org/coar/resource_type/c_6501info:ar-repo/semantics/articuloapplication/pdfapplication/pdfhttp://hdl.handle.net/11336/42845Alvarez, Nicolás Alejandro; Becher, Veronica Andrea; M. Levin’s construction of absolutely normal numbers with very low discrepancy; American Mathematical Society; Mathematics Of Computation; 86; 3-2017; 2927-29460025-57181088-6842CONICET DigitalCONICETenginfo:eu-repo/semantics/altIdentifier/url/http://www.ams.org/journals/mcom/2017-86-308/S0025-5718-2017-03188-4/info:eu-repo/semantics/altIdentifier/doi/10.1090/mcom/3188info:eu-repo/semantics/altIdentifier/url/https://arxiv.org/abs/1510.02004info:eu-repo/semantics/openAccesshttps://creativecommons.org/licenses/by-nc-nd/2.5/ar/reponame:CONICET Digital (CONICET)instname:Consejo Nacional de Investigaciones Científicas y Técnicas2024-05-08T14:13:51Zoai:ri.conicet.gov.ar:11336/42845instacron:CONICETInstitucionalhttp://ri.conicet.gov.ar/Organismo científico-tecnológicoNo correspondehttp://ri.conicet.gov.ar/oai/requestdasensio@conicet.gov.ar; lcarlino@conicet.gov.arArgentinaNo correspondeNo correspondeNo correspondeopendoar:34982024-05-08 14:13:51.866CONICET Digital (CONICET) - Consejo Nacional de Investigaciones Científicas y Técnicasfalse
dc.title.none.fl_str_mv M. Levin’s construction of absolutely normal numbers with very low discrepancy
title M. Levin’s construction of absolutely normal numbers with very low discrepancy
spellingShingle M. Levin’s construction of absolutely normal numbers with very low discrepancy
Alvarez, Nicolás Alejandro
NORMAL NUMBERS
DISCREPANCY
ALGORITHMS
https://purl.org/becyt/ford/1.2
https://purl.org/becyt/ford/1
title_short M. Levin’s construction of absolutely normal numbers with very low discrepancy
title_full M. Levin’s construction of absolutely normal numbers with very low discrepancy
title_fullStr M. Levin’s construction of absolutely normal numbers with very low discrepancy
title_full_unstemmed M. Levin’s construction of absolutely normal numbers with very low discrepancy
title_sort M. Levin’s construction of absolutely normal numbers with very low discrepancy
dc.creator.none.fl_str_mv Alvarez, Nicolás Alejandro
Becher, Veronica Andrea
author Alvarez, Nicolás Alejandro
author_facet Alvarez, Nicolás Alejandro
Becher, Veronica Andrea
author_role author
author2 Becher, Veronica Andrea
author2_role author
dc.subject.none.fl_str_mv NORMAL NUMBERS
DISCREPANCY
ALGORITHMS
https://purl.org/becyt/ford/1.2
https://purl.org/becyt/ford/1
topic NORMAL NUMBERS
DISCREPANCY
ALGORITHMS
https://purl.org/becyt/ford/1.2
https://purl.org/becyt/ford/1
description Among the currently known constructions of absolutely normal numbers, the one given by Mordechay Levin in 1979 achieves the lowest discrepancy bound. In this work we analyze this construction in terms of computability and computational complexity. We show that, under basic assumptions, it yields a computable real number. The construction does not give the digits of the fractional expansion explicitly, but it gives a sequence of increasing approximations whose limit is the announced absolutely normal number. The nth approximation has an error less than 2–2n. To obtain the $ nth approximation the construction requires, in the worst case, a number of mathematical operations that is doubly exponential in n. We consider variants on the construction that reduce the computational complexity at the expense of an increment in discrepancy.
publishDate 2017
dc.date.none.fl_str_mv 2017-03
dc.type.none.fl_str_mv info:eu-repo/semantics/article
info:eu-repo/semantics/publishedVersion
http://purl.org/coar/resource_type/c_6501
info:ar-repo/semantics/articulo
format article
status_str publishedVersion
dc.identifier.none.fl_str_mv http://hdl.handle.net/11336/42845
Alvarez, Nicolás Alejandro; Becher, Veronica Andrea; M. Levin’s construction of absolutely normal numbers with very low discrepancy; American Mathematical Society; Mathematics Of Computation; 86; 3-2017; 2927-2946
0025-5718
1088-6842
CONICET Digital
CONICET
url http://hdl.handle.net/11336/42845
identifier_str_mv Alvarez, Nicolás Alejandro; Becher, Veronica Andrea; M. Levin’s construction of absolutely normal numbers with very low discrepancy; American Mathematical Society; Mathematics Of Computation; 86; 3-2017; 2927-2946
0025-5718
1088-6842
CONICET Digital
CONICET
dc.language.none.fl_str_mv eng
language eng
dc.relation.none.fl_str_mv info:eu-repo/semantics/altIdentifier/url/http://www.ams.org/journals/mcom/2017-86-308/S0025-5718-2017-03188-4/
info:eu-repo/semantics/altIdentifier/doi/10.1090/mcom/3188
info:eu-repo/semantics/altIdentifier/url/https://arxiv.org/abs/1510.02004
dc.rights.none.fl_str_mv info:eu-repo/semantics/openAccess
https://creativecommons.org/licenses/by-nc-nd/2.5/ar/
eu_rights_str_mv openAccess
rights_invalid_str_mv https://creativecommons.org/licenses/by-nc-nd/2.5/ar/
dc.format.none.fl_str_mv application/pdf
application/pdf
dc.publisher.none.fl_str_mv American Mathematical Society
publisher.none.fl_str_mv American Mathematical Society
dc.source.none.fl_str_mv reponame:CONICET Digital (CONICET)
instname:Consejo Nacional de Investigaciones Científicas y Técnicas
instname_str Consejo Nacional de Investigaciones Científicas y Técnicas
reponame_str CONICET Digital (CONICET)
collection CONICET Digital (CONICET)
repository.name.fl_str_mv CONICET Digital (CONICET) - Consejo Nacional de Investigaciones Científicas y Técnicas
repository.mail.fl_str_mv dasensio@conicet.gov.ar; lcarlino@conicet.gov.ar
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score 15.812429

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