On simply normal numbers to different bases
- Autores
- Becher, Veronica Andrea; Bugeaud, Yann; Slaman, Theodore A.
- Año de publicación
- 2016
- Idioma
- inglés
- Tipo de recurso
- artículo
- Estado
- versión publicada
- Descripción
- Let s be an integer greater than or equal to 2. A real number is simply normal to base s if in its base-s expansion every digit 0,1,…,s-1 occurs with the same frequency 1/s. Let S be the set of positive integers that are not perfect powers, hence S is the set {2,3,5,6,7,10,11,…}. Let M be a function from S to sets of positive integers such that, for each s in S, if m is in M(s) then each divisor of m is in M(s) and if M(s) is infinite then it is equal to the set of all positive integers. These conditions on M are necessary for there to be a real number which is simply normal to exactly the bases sm such that s is in S and m is in M(s). We show these conditions are also sufficient and further establish that the set of real numbers that satisfy them has full Hausdorff dimension. This extends a result of W. M. Schmidt (1961/1962) on normal numbers to different bases.
Fil: Becher, Veronica Andrea. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigación en Ciencias de la Computación. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigación en Ciencias de la Computación; Argentina
Fil: Bugeaud, Yann. Université de Strasbourg; Francia
Fil: Slaman, Theodore A.. University of California at Berkeley; Estados Unidos - Materia
-
Normal Numbers
Simply Normal Numbers - 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:11336/60110
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On simply normal numbers to different basesBecher, Veronica AndreaBugeaud, YannSlaman, Theodore A.Normal NumbersSimply Normal Numbershttps://purl.org/becyt/ford/1.2https://purl.org/becyt/ford/1Let s be an integer greater than or equal to 2. A real number is simply normal to base s if in its base-s expansion every digit 0,1,…,s-1 occurs with the same frequency 1/s. Let S be the set of positive integers that are not perfect powers, hence S is the set {2,3,5,6,7,10,11,…}. Let M be a function from S to sets of positive integers such that, for each s in S, if m is in M(s) then each divisor of m is in M(s) and if M(s) is infinite then it is equal to the set of all positive integers. These conditions on M are necessary for there to be a real number which is simply normal to exactly the bases sm such that s is in S and m is in M(s). We show these conditions are also sufficient and further establish that the set of real numbers that satisfy them has full Hausdorff dimension. This extends a result of W. M. Schmidt (1961/1962) on normal numbers to different bases.Fil: Becher, Veronica Andrea. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigación en Ciencias de la Computación. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigación en Ciencias de la Computación; ArgentinaFil: Bugeaud, Yann. Université de Strasbourg; FranciaFil: Slaman, Theodore A.. University of California at Berkeley; Estados UnidosSpringer2016-02info:eu-repo/semantics/articleinfo:eu-repo/semantics/publishedVersionhttp://purl.org/coar/resource_type/c_6501info:ar-repo/semantics/articuloapplication/pdfapplication/pdfapplication/pdfhttp://hdl.handle.net/11336/60110Becher, Veronica Andrea; Bugeaud, Yann; Slaman, Theodore A.; On simply normal numbers to different bases; Springer; Mathematische Annalen; 364; 1-2; 2-2016; 125-1500025-5831CONICET DigitalCONICETenginfo:eu-repo/semantics/altIdentifier/doi/10.1007/s00208-015-1209-9info:eu-repo/semantics/altIdentifier/url/https://link.springer.com/article/10.1007/s00208-015-1209-9info:eu-repo/semantics/altIdentifier/url/https://arxiv.org/abs/1311.0332info:eu-repo/semantics/openAccesshttps://creativecommons.org/licenses/by-nc-nd/2.5/ar/reponame:CONICET Digital (CONICET)instname:Consejo Nacional de Investigaciones Científicas y Técnicas2025-09-03T09:44:51Zoai:ri.conicet.gov.ar:11336/60110instacron: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 09:44:51.544CONICET Digital (CONICET) - Consejo Nacional de Investigaciones Científicas y Técnicasfalse |
dc.title.none.fl_str_mv |
On simply normal numbers to different bases |
title |
On simply normal numbers to different bases |
spellingShingle |
On simply normal numbers to different bases Becher, Veronica Andrea Normal Numbers Simply Normal Numbers |
title_short |
On simply normal numbers to different bases |
title_full |
On simply normal numbers to different bases |
title_fullStr |
On simply normal numbers to different bases |
title_full_unstemmed |
On simply normal numbers to different bases |
title_sort |
On simply normal numbers to different bases |
dc.creator.none.fl_str_mv |
Becher, Veronica Andrea Bugeaud, Yann Slaman, Theodore A. |
author |
Becher, Veronica Andrea |
author_facet |
Becher, Veronica Andrea Bugeaud, Yann Slaman, Theodore A. |
author_role |
author |
author2 |
Bugeaud, Yann Slaman, Theodore A. |
author2_role |
author author |
dc.subject.none.fl_str_mv |
Normal Numbers Simply Normal Numbers |
topic |
Normal Numbers Simply Normal Numbers |
purl_subject.fl_str_mv |
https://purl.org/becyt/ford/1.2 https://purl.org/becyt/ford/1 |
dc.description.none.fl_txt_mv |
Let s be an integer greater than or equal to 2. A real number is simply normal to base s if in its base-s expansion every digit 0,1,…,s-1 occurs with the same frequency 1/s. Let S be the set of positive integers that are not perfect powers, hence S is the set {2,3,5,6,7,10,11,…}. Let M be a function from S to sets of positive integers such that, for each s in S, if m is in M(s) then each divisor of m is in M(s) and if M(s) is infinite then it is equal to the set of all positive integers. These conditions on M are necessary for there to be a real number which is simply normal to exactly the bases sm such that s is in S and m is in M(s). We show these conditions are also sufficient and further establish that the set of real numbers that satisfy them has full Hausdorff dimension. This extends a result of W. M. Schmidt (1961/1962) on normal numbers to different bases. Fil: Becher, Veronica Andrea. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigación en Ciencias de la Computación. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigación en Ciencias de la Computación; Argentina Fil: Bugeaud, Yann. Université de Strasbourg; Francia Fil: Slaman, Theodore A.. University of California at Berkeley; Estados Unidos |
description |
Let s be an integer greater than or equal to 2. A real number is simply normal to base s if in its base-s expansion every digit 0,1,…,s-1 occurs with the same frequency 1/s. Let S be the set of positive integers that are not perfect powers, hence S is the set {2,3,5,6,7,10,11,…}. Let M be a function from S to sets of positive integers such that, for each s in S, if m is in M(s) then each divisor of m is in M(s) and if M(s) is infinite then it is equal to the set of all positive integers. These conditions on M are necessary for there to be a real number which is simply normal to exactly the bases sm such that s is in S and m is in M(s). We show these conditions are also sufficient and further establish that the set of real numbers that satisfy them has full Hausdorff dimension. This extends a result of W. M. Schmidt (1961/1962) on normal numbers to different bases. |
publishDate |
2016 |
dc.date.none.fl_str_mv |
2016-02 |
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/11336/60110 Becher, Veronica Andrea; Bugeaud, Yann; Slaman, Theodore A.; On simply normal numbers to different bases; Springer; Mathematische Annalen; 364; 1-2; 2-2016; 125-150 0025-5831 CONICET Digital CONICET |
url |
http://hdl.handle.net/11336/60110 |
identifier_str_mv |
Becher, Veronica Andrea; Bugeaud, Yann; Slaman, Theodore A.; On simply normal numbers to different bases; Springer; Mathematische Annalen; 364; 1-2; 2-2016; 125-150 0025-5831 CONICET Digital CONICET |
dc.language.none.fl_str_mv |
eng |
language |
eng |
dc.relation.none.fl_str_mv |
info:eu-repo/semantics/altIdentifier/doi/10.1007/s00208-015-1209-9 info:eu-repo/semantics/altIdentifier/url/https://link.springer.com/article/10.1007/s00208-015-1209-9 info:eu-repo/semantics/altIdentifier/url/https://arxiv.org/abs/1311.0332 |
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 application/pdf |
dc.publisher.none.fl_str_mv |
Springer |
publisher.none.fl_str_mv |
Springer |
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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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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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