Classical and uniform exponents of multiplicative p -adic approximation
Let p be a prime number and ξ an irrational p-adic number. Its irrationality exponent µ(ξ) is the supremum of the real numbers µ for which the system of inequalities 0 < max{|x|, |y|} ≤ X, |yξ - x|p ≤ X-µ has a solution in integers x, y for arbitrarily large real number X. Its multiplicative irra...
| Autores: | , |
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| Tipo de recurso: | artículo |
| Fecha de publicación: | 2024 |
| País: | España |
| Institución: | Universitat Autònoma de Barcelona |
| Repositorio: | Dipòsit Digital de Documents de la UAB |
| Idioma: | inglés |
| OAI Identifier: | oai:ddd.uab.cat:286796 |
| Acceso en línea: | https://ddd.uab.cat/record/286796 https://dx.doi.org/urn:doi:10.5565/PUBLMAT6812401 |
| Access Level: | acceso abierto |
| Palabra clave: | Rational approximation P-adic number Exponent of approximation |
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Classical and uniform exponents of multiplicative p -adic approximationBugeaud, YannSchleischitz, JohannesRational approximationP-adic numberExponent of approximationLet p be a prime number and ξ an irrational p-adic number. Its irrationality exponent µ(ξ) is the supremum of the real numbers µ for which the system of inequalities 0 < max{|x|, |y|} ≤ X, |yξ - x|p ≤ X-µ has a solution in integers x, y for arbitrarily large real number X. Its multiplicative irrationality exponent µ×(ξ) (resp., uniform multiplicative irrationality exponent µb×(ξ)) is the supremum of the real numbers µb for which the system of inequalities 0 < |xy| 1/2 ≤ X, |yξ - x|p ≤ X-µb has a solution in integers x, y for arbitrarily large (resp., for every sufficiently large) real number X. It is not difficult to show that µ(ξ) ≤ µ×(ξ) ≤ 2µ(ξ) and µb×(ξ) ≤ 4. We establish that the ratio between the multiplicative irrationality exponent µ× and the irrationality exponent µ can take any given value in [1, 2]. Furthermore, we prove that µb×(ξ) ≤ (5 + √ 5)/2 for every p-adic number ξ. 22024-01-0120242024-01-01Articlehttp://purl.org/coar/resource_type/c_6501VoRhttp://purl.org/coar/version/c_970fb48d4fbd8a85info:eu-repo/semantics/articleapplication/pdfhttps://ddd.uab.cat/record/286796https://dx.doi.org/urn:doi:10.5565/PUBLMAT6812401reponame:Dipòsit Digital de Documents de la UABinstname:Universitat Autònoma de BarcelonaInglésengopen accesshttp://purl.org/coar/access_right/c_abf2Aquest material està protegit per drets d'autor i/o drets afins. Podeu utilitzar aquest material en funció del que permet la legislació de drets d'autor i drets afins d'aplicació al vostre cas. Per a d'altres usos heu d'obtenir permís del(s) titular(s) de drets.https://rightsstatements.org/vocab/InC/1.0/info:eu-repo/semantics/openAccessoai:ddd.uab.cat:2867962026-06-06T12:50:31Z |
| dc.title.none.fl_str_mv |
Classical and uniform exponents of multiplicative p -adic approximation |
| title |
Classical and uniform exponents of multiplicative p -adic approximation |
| spellingShingle |
Classical and uniform exponents of multiplicative p -adic approximation Bugeaud, Yann Rational approximation P-adic number Exponent of approximation |
| title_short |
Classical and uniform exponents of multiplicative p -adic approximation |
| title_full |
Classical and uniform exponents of multiplicative p -adic approximation |
| title_fullStr |
Classical and uniform exponents of multiplicative p -adic approximation |
| title_full_unstemmed |
Classical and uniform exponents of multiplicative p -adic approximation |
| title_sort |
Classical and uniform exponents of multiplicative p -adic approximation |
| dc.creator.none.fl_str_mv |
Bugeaud, Yann Schleischitz, Johannes |
| author |
Bugeaud, Yann |
| author_facet |
Bugeaud, Yann Schleischitz, Johannes |
| author_role |
author |
| author2 |
Schleischitz, Johannes |
| author2_role |
author |
| dc.subject.none.fl_str_mv |
Rational approximation P-adic number Exponent of approximation |
| topic |
Rational approximation P-adic number Exponent of approximation |
| description |
Let p be a prime number and ξ an irrational p-adic number. Its irrationality exponent µ(ξ) is the supremum of the real numbers µ for which the system of inequalities 0 < max{|x|, |y|} ≤ X, |yξ - x|p ≤ X-µ has a solution in integers x, y for arbitrarily large real number X. Its multiplicative irrationality exponent µ×(ξ) (resp., uniform multiplicative irrationality exponent µb×(ξ)) is the supremum of the real numbers µb for which the system of inequalities 0 < |xy| 1/2 ≤ X, |yξ - x|p ≤ X-µb has a solution in integers x, y for arbitrarily large (resp., for every sufficiently large) real number X. It is not difficult to show that µ(ξ) ≤ µ×(ξ) ≤ 2µ(ξ) and µb×(ξ) ≤ 4. We establish that the ratio between the multiplicative irrationality exponent µ× and the irrationality exponent µ can take any given value in [1, 2]. Furthermore, we prove that µb×(ξ) ≤ (5 + √ 5)/2 for every p-adic number ξ. |
| publishDate |
2024 |
| dc.date.none.fl_str_mv |
2 2024-01-01 2024 2024-01-01 |
| dc.type.none.fl_str_mv |
Article http://purl.org/coar/resource_type/c_6501 VoR http://purl.org/coar/version/c_970fb48d4fbd8a85 |
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info:eu-repo/semantics/article |
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article |
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https://ddd.uab.cat/record/286796 https://dx.doi.org/urn:doi:10.5565/PUBLMAT6812401 |
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https://ddd.uab.cat/record/286796 https://dx.doi.org/urn:doi:10.5565/PUBLMAT6812401 |
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Inglés eng |
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Inglés |
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eng |
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open access http://purl.org/coar/access_right/c_abf2 https://rightsstatements.org/vocab/InC/1.0/ |
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info:eu-repo/semantics/openAccess |
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open access http://purl.org/coar/access_right/c_abf2 https://rightsstatements.org/vocab/InC/1.0/ |
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openAccess |
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application/pdf |
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reponame:Dipòsit Digital de Documents de la UAB instname:Universitat Autònoma de Barcelona |
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Universitat Autònoma de Barcelona |
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