On the normality of numbers to different bases

Autores
Becher, Veronica Andrea; Slaman, Theodore A.
Año de publicación
2014
Idioma
inglés
Tipo de recurso
artículo
Estado
versión publicada
Descripción
We demonstrate the full logical independence of normality to multiplicatively independent bases. This establishes that the set of bases to which a real number can be normal is not tied to any arithmetical properties other than multiplicative dependence. It also establishes that the set of real numbers which are normal to at least one base is properly at the fourth level of the Borel hierarchy, which was conjectured by A. Ditzen 20 years ago. We further show that the discrepancy functions for multiplicatively independent bases are pairwise independent. In addition, for any given set of bases closed under multiplicative dependence, there are real numbers that are normal to each base in the given set, but not simply normal to any base in its complement. This answers a question first raised by Brown, Moran and Pearce.
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
Fil: Slaman, Theodore A.. University of California at Berkeley; Estados Unidos
Materia
Normal Numbers
Descriptive Set Theory
Normality to Different Bases
Discrepancy
Nivel de accesibilidad
acceso abierto
Condiciones de uso
https://creativecommons.org/licenses/by-nc-sa/2.5/ar/
Repositorio
CONICET Digital (CONICET)
Institución
Consejo Nacional de Investigaciones Científicas y Técnicas
OAI Identificador
oai:ri.conicet.gov.ar:11336/33088
Acceder
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oai_identifier_str oai:ri.conicet.gov.ar:11336/33088
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network_name_str CONICET Digital (CONICET)
spelling On the normality of numbers to different basesBecher, Veronica AndreaSlaman, Theodore A.Normal NumbersDescriptive Set TheoryNormality to Different BasesDiscrepancyhttps://purl.org/becyt/ford/1.2https://purl.org/becyt/ford/1We demonstrate the full logical independence of normality to multiplicatively independent bases. This establishes that the set of bases to which a real number can be normal is not tied to any arithmetical properties other than multiplicative dependence. It also establishes that the set of real numbers which are normal to at least one base is properly at the fourth level of the Borel hierarchy, which was conjectured by A. Ditzen 20 years ago. We further show that the discrepancy functions for multiplicatively independent bases are pairwise independent. In addition, for any given set of bases closed under multiplicative dependence, there are real numbers that are normal to each base in the given set, but not simply normal to any base in its complement. This answers a question first raised by Brown, Moran and Pearce.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; ArgentinaFil: Slaman, Theodore A.. University of California at Berkeley; Estados UnidosWiley2014-07info:eu-repo/semantics/articleinfo:eu-repo/semantics/publishedVersionhttp://purl.org/coar/resource_type/c_6501info:ar-repo/semantics/articuloapplication/pdfapplication/pdfhttp://hdl.handle.net/11336/33088Becher, Veronica Andrea; Slaman, Theodore A.; On the normality of numbers to different bases; Wiley; Proceedings of the London Mathematical Society; 90; 2; 7-2014; 472-4940024-6115CONICET DigitalCONICETenginfo:eu-repo/semantics/altIdentifier/doi/10.1112/jlms/jdu035info:eu-repo/semantics/altIdentifier/url/http://onlinelibrary.wiley.com/doi/10.1112/jlms/jdu035/abstractinfo:eu-repo/semantics/openAccesshttps://creativecommons.org/licenses/by-nc-sa/2.5/ar/reponame:CONICET Digital (CONICET)instname:Consejo Nacional de Investigaciones Científicas y Técnicas2025-09-29T09:44:47Zoai:ri.conicet.gov.ar:11336/33088instacron: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:34982025-09-29 09:44:47.928CONICET Digital (CONICET) - Consejo Nacional de Investigaciones Científicas y Técnicasfalse
dc.title.none.fl_str_mv On the normality of numbers to different bases
title On the normality of numbers to different bases
spellingShingle On the normality of numbers to different bases
Becher, Veronica Andrea
Normal Numbers
Descriptive Set Theory
Normality to Different Bases
Discrepancy
title_short On the normality of numbers to different bases
title_full On the normality of numbers to different bases
title_fullStr On the normality of numbers to different bases
title_full_unstemmed On the normality of numbers to different bases
title_sort On the normality of numbers to different bases
dc.creator.none.fl_str_mv Becher, Veronica Andrea
Slaman, Theodore A.
author Becher, Veronica Andrea
author_facet Becher, Veronica Andrea
Slaman, Theodore A.
author_role author
author2 Slaman, Theodore A.
author2_role author
dc.subject.none.fl_str_mv Normal Numbers
Descriptive Set Theory
Normality to Different Bases
Discrepancy
topic Normal Numbers
Descriptive Set Theory
Normality to Different Bases
Discrepancy
purl_subject.fl_str_mv https://purl.org/becyt/ford/1.2
https://purl.org/becyt/ford/1
dc.description.none.fl_txt_mv We demonstrate the full logical independence of normality to multiplicatively independent bases. This establishes that the set of bases to which a real number can be normal is not tied to any arithmetical properties other than multiplicative dependence. It also establishes that the set of real numbers which are normal to at least one base is properly at the fourth level of the Borel hierarchy, which was conjectured by A. Ditzen 20 years ago. We further show that the discrepancy functions for multiplicatively independent bases are pairwise independent. In addition, for any given set of bases closed under multiplicative dependence, there are real numbers that are normal to each base in the given set, but not simply normal to any base in its complement. This answers a question first raised by Brown, Moran and Pearce.
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
Fil: Slaman, Theodore A.. University of California at Berkeley; Estados Unidos
description We demonstrate the full logical independence of normality to multiplicatively independent bases. This establishes that the set of bases to which a real number can be normal is not tied to any arithmetical properties other than multiplicative dependence. It also establishes that the set of real numbers which are normal to at least one base is properly at the fourth level of the Borel hierarchy, which was conjectured by A. Ditzen 20 years ago. We further show that the discrepancy functions for multiplicatively independent bases are pairwise independent. In addition, for any given set of bases closed under multiplicative dependence, there are real numbers that are normal to each base in the given set, but not simply normal to any base in its complement. This answers a question first raised by Brown, Moran and Pearce.
publishDate 2014
dc.date.none.fl_str_mv 2014-07
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/11336/33088
Becher, Veronica Andrea; Slaman, Theodore A.; On the normality of numbers to different bases; Wiley; Proceedings of the London Mathematical Society; 90; 2; 7-2014; 472-494
0024-6115
CONICET Digital
CONICET
url http://hdl.handle.net/11336/33088
identifier_str_mv Becher, Veronica Andrea; Slaman, Theodore A.; On the normality of numbers to different bases; Wiley; Proceedings of the London Mathematical Society; 90; 2; 7-2014; 472-494
0024-6115
CONICET Digital
CONICET
dc.language.none.fl_str_mv eng
language eng
dc.relation.none.fl_str_mv info:eu-repo/semantics/altIdentifier/doi/10.1112/jlms/jdu035
info:eu-repo/semantics/altIdentifier/url/http://onlinelibrary.wiley.com/doi/10.1112/jlms/jdu035/abstract
dc.rights.none.fl_str_mv info:eu-repo/semantics/openAccess
https://creativecommons.org/licenses/by-nc-sa/2.5/ar/
eu_rights_str_mv openAccess
rights_invalid_str_mv https://creativecommons.org/licenses/by-nc-sa/2.5/ar/
dc.format.none.fl_str_mv application/pdf
application/pdf
dc.publisher.none.fl_str_mv Wiley
publisher.none.fl_str_mv Wiley
dc.source.none.fl_str_mv reponame:CONICET Digital (CONICET)
instname:Consejo Nacional de Investigaciones Científicas y Técnicas
reponame_str CONICET Digital (CONICET)
collection CONICET Digital (CONICET)
instname_str Consejo Nacional de Investigaciones Científicas y Técnicas
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
_version_ 1844613409641857024
score 13.069144

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