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
- Institución
- Consejo Nacional de Investigaciones Científicas y Técnicas
- OAI Identificador
- oai:ri.conicet.gov.ar:11336/33088
Ver los metadatos del registro completo
id |
CONICETDig_be541e56337f057f8b0a8d3168d294df |
---|---|
oai_identifier_str |
oai:ri.conicet.gov.ar:11336/33088 |
network_acronym_str |
CONICETDig |
repository_id_str |
3498 |
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 Técnicas2025-09-03T09:50:11Zoai: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:34982025-09-03 09:50:11.433CONICET 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_ |
1842269018599194624 |
score |
13.13397 |