Irrationality exponent, Hausdorff dimension and effectivization

Autores
Becher, Veronica Andrea; Reimann, Jan; Slaman, Theodore A.
Año de publicación
2018
Idioma
inglés
Tipo de recurso
artículo
Estado
versión publicada
Descripción
We generalize the classical theorem by Jarnik and Besicovitch on the irrationality exponents of real numbers and Hausdorff dimension and show that the two notions are independent. For any real number a greater than or equal to 2 and any non-negative real b be less than or equal to 2 / a, we show that there is a Cantor-like set with Hausdorff dimension equal to b such that, with respect to its uniform measure, almost all real numbers have irrationality exponent equal to a. We give an analogous result relating the irrationality exponent and the effective Hausdorff dimension of individual real numbers. We prove that there is a Cantor-like set such that, with respect to its uniform measure, almost all elements in the set have effective Hausdorff dimension equal to b and irrationality exponent equal to a. In each case, we obtain the desired set as a distinguished path in a tree of Cantor sets.
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. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Departamento de Computación; Argentina
Fil: Reimann, Jan. State University of Pennsylvania; Estados Unidos
Fil: Slaman, Theodore A.. University of California. Department of Mathematics; Estados Unidos
Materia
CANTOR SETS
DIOPHANTINE APPROXIMATION
EFFECTIVE HAUSDORFF DIMENSION
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:11336/97123

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spelling Irrationality exponent, Hausdorff dimension and effectivizationBecher, Veronica AndreaReimann, JanSlaman, Theodore A.CANTOR SETSDIOPHANTINE APPROXIMATIONEFFECTIVE HAUSDORFF DIMENSIONhttps://purl.org/becyt/ford/1.2https://purl.org/becyt/ford/1We generalize the classical theorem by Jarnik and Besicovitch on the irrationality exponents of real numbers and Hausdorff dimension and show that the two notions are independent. For any real number a greater than or equal to 2 and any non-negative real b be less than or equal to 2 / a, we show that there is a Cantor-like set with Hausdorff dimension equal to b such that, with respect to its uniform measure, almost all real numbers have irrationality exponent equal to a. We give an analogous result relating the irrationality exponent and the effective Hausdorff dimension of individual real numbers. We prove that there is a Cantor-like set such that, with respect to its uniform measure, almost all elements in the set have effective Hausdorff dimension equal to b and irrationality exponent equal to a. In each case, we obtain the desired set as a distinguished path in a tree of Cantor sets.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. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Departamento de Computación; ArgentinaFil: Reimann, Jan. State University of Pennsylvania; Estados UnidosFil: Slaman, Theodore A.. University of California. Department of Mathematics; Estados UnidosSpringer Wien2018-02info: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/11336/97123Becher, Veronica Andrea; Reimann, Jan; Slaman, Theodore A.; Irrationality exponent, Hausdorff dimension and effectivization; Springer Wien; Monatshefete Fur Mathematik; 185; 2; 2-2018; 167-1880026-9255CONICET DigitalCONICETenginfo:eu-repo/semantics/altIdentifier/url/https://link.springer.com/article/10.1007%2Fs00605-017-1094-2info:eu-repo/semantics/altIdentifier/doi/10.1007/s00605-017-1094-2info:eu-repo/semantics/altIdentifier/url/https://arxiv.org/abs/1601.00153info:eu-repo/semantics/openAccesshttps://creativecommons.org/licenses/by-nc-nd/2.5/ar/reponame:CONICET Digital (CONICET)instname:Consejo Nacional de Investigaciones Científicas y Técnicas2025-09-03T09:49:59Zoai:ri.conicet.gov.ar:11336/97123instacron: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 09:49:59.66CONICET Digital (CONICET) - Consejo Nacional de Investigaciones Científicas y Técnicasfalse
dc.title.none.fl_str_mv Irrationality exponent, Hausdorff dimension and effectivization
title Irrationality exponent, Hausdorff dimension and effectivization
spellingShingle Irrationality exponent, Hausdorff dimension and effectivization
Becher, Veronica Andrea
CANTOR SETS
DIOPHANTINE APPROXIMATION
EFFECTIVE HAUSDORFF DIMENSION
title_short Irrationality exponent, Hausdorff dimension and effectivization
title_full Irrationality exponent, Hausdorff dimension and effectivization
title_fullStr Irrationality exponent, Hausdorff dimension and effectivization
title_full_unstemmed Irrationality exponent, Hausdorff dimension and effectivization
title_sort Irrationality exponent, Hausdorff dimension and effectivization
dc.creator.none.fl_str_mv Becher, Veronica Andrea
Reimann, Jan
Slaman, Theodore A.
author Becher, Veronica Andrea
author_facet Becher, Veronica Andrea
Reimann, Jan
Slaman, Theodore A.
author_role author
author2 Reimann, Jan
Slaman, Theodore A.
author2_role author
author
dc.subject.none.fl_str_mv CANTOR SETS
DIOPHANTINE APPROXIMATION
EFFECTIVE HAUSDORFF DIMENSION
topic CANTOR SETS
DIOPHANTINE APPROXIMATION
EFFECTIVE HAUSDORFF DIMENSION
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 generalize the classical theorem by Jarnik and Besicovitch on the irrationality exponents of real numbers and Hausdorff dimension and show that the two notions are independent. For any real number a greater than or equal to 2 and any non-negative real b be less than or equal to 2 / a, we show that there is a Cantor-like set with Hausdorff dimension equal to b such that, with respect to its uniform measure, almost all real numbers have irrationality exponent equal to a. We give an analogous result relating the irrationality exponent and the effective Hausdorff dimension of individual real numbers. We prove that there is a Cantor-like set such that, with respect to its uniform measure, almost all elements in the set have effective Hausdorff dimension equal to b and irrationality exponent equal to a. In each case, we obtain the desired set as a distinguished path in a tree of Cantor sets.
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. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Departamento de Computación; Argentina
Fil: Reimann, Jan. State University of Pennsylvania; Estados Unidos
Fil: Slaman, Theodore A.. University of California. Department of Mathematics; Estados Unidos
description We generalize the classical theorem by Jarnik and Besicovitch on the irrationality exponents of real numbers and Hausdorff dimension and show that the two notions are independent. For any real number a greater than or equal to 2 and any non-negative real b be less than or equal to 2 / a, we show that there is a Cantor-like set with Hausdorff dimension equal to b such that, with respect to its uniform measure, almost all real numbers have irrationality exponent equal to a. We give an analogous result relating the irrationality exponent and the effective Hausdorff dimension of individual real numbers. We prove that there is a Cantor-like set such that, with respect to its uniform measure, almost all elements in the set have effective Hausdorff dimension equal to b and irrationality exponent equal to a. In each case, we obtain the desired set as a distinguished path in a tree of Cantor sets.
publishDate 2018
dc.date.none.fl_str_mv 2018-02
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/11336/97123
Becher, Veronica Andrea; Reimann, Jan; Slaman, Theodore A.; Irrationality exponent, Hausdorff dimension and effectivization; Springer Wien; Monatshefete Fur Mathematik; 185; 2; 2-2018; 167-188
0026-9255
CONICET Digital
CONICET
url http://hdl.handle.net/11336/97123
identifier_str_mv Becher, Veronica Andrea; Reimann, Jan; Slaman, Theodore A.; Irrationality exponent, Hausdorff dimension and effectivization; Springer Wien; Monatshefete Fur Mathematik; 185; 2; 2-2018; 167-188
0026-9255
CONICET Digital
CONICET
dc.language.none.fl_str_mv eng
language eng
dc.relation.none.fl_str_mv info:eu-repo/semantics/altIdentifier/url/https://link.springer.com/article/10.1007%2Fs00605-017-1094-2
info:eu-repo/semantics/altIdentifier/doi/10.1007/s00605-017-1094-2
info:eu-repo/semantics/altIdentifier/url/https://arxiv.org/abs/1601.00153
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 Wien
publisher.none.fl_str_mv Springer Wien
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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