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
CONICET Digital (CONICET)
Institución
Consejo Nacional de Investigaciones Científicas y Técnicas
OAI Identificador
oai:ri.conicet.gov.ar:11336/60110
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network_name_str CONICET Digital (CONICET)
spelling 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 Técnicas2025-09-29T09:35:00Zoai: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:34982025-09-29 09:35:00.818CONICET 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
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
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score 13.070432

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