M. Levin’s construction of absolutely normal numbers with very low discrepancy
- Autores
- Alvarez, Nicolás Alejandro; Becher, Veronica Andrea
- Año de publicación
- 2017
- Idioma
- inglés
- Tipo de recurso
- artículo
- Estado
- versión publicada
- Descripción
- 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.
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; Argentina
Fil: Becher, Veronica Andrea. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Departamento de Computación; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina - Materia
-
NORMAL NUMBERS
DISCREPANCY
ALGORITHMS - Nivel de accesibilidad
- acceso abierto
- Condiciones de uso
- https://creativecommons.org/licenses/by-nc-nd/2.5/ar/
- Repositorio
- Institución
- Consejo Nacional de Investigaciones Científicas y Técnicas
- OAI Identificador
- oai:ri.conicet.gov.ar:11336/42845
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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 Técnicas2025-09-03T10:05:37Zoai: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:34982025-09-03 10:05:37.701CONICET 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 |
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 |
topic |
NORMAL NUMBERS DISCREPANCY ALGORITHMS |
purl_subject.fl_str_mv |
https://purl.org/becyt/ford/1.2 https://purl.org/becyt/ford/1 |
dc.description.none.fl_txt_mv |
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. 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; Argentina Fil: Becher, Veronica Andrea. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Departamento de Computación; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina |
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 |
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reponame:CONICET Digital (CONICET) instname:Consejo Nacional de Investigaciones Científicas y Técnicas |
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CONICET Digital (CONICET) |
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CONICET Digital (CONICET) |
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Consejo Nacional de Investigaciones Científicas y Técnicas |
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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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13.13397 |