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
- Institución
- Consejo Nacional de Investigaciones Científicas y Técnicas
- OAI Identificador
- oai:ri.conicet.gov.ar:11336/97123
Ver los metadatos del registro completo
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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/ |
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openAccess |
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https://creativecommons.org/licenses/by-nc-nd/2.5/ar/ |
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Springer Wien |
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Springer Wien |
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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 |
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dasensio@conicet.gov.ar; lcarlino@conicet.gov.ar |
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