Irrationality exponent, Hausdorff dimension and effectivization
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 tha...
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| Acceso en línea: | http://hdl.handle.net/20.500.12110/paper_00269255_v185_n2_p167_Becher |
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todo:paper_00269255_v185_n2_p167_Becher2023-10-03T14:37:23Z Irrationality exponent, Hausdorff dimension and effectivization Becher, V. Reimann, J. Slaman, T.A. Cantor sets Diophantine approximation Effective Hausdorff dimension 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. © 2017, Springer-Verlag GmbH Austria. JOUR info:eu-repo/semantics/openAccess http://creativecommons.org/licenses/by/2.5/ar http://hdl.handle.net/20.500.12110/paper_00269255_v185_n2_p167_Becher |
| institution |
Universidad de Buenos Aires |
| institution_str |
I-28 |
| repository_str |
R-134 |
| collection |
Biblioteca Digital - Facultad de Ciencias Exactas y Naturales (UBA) |
| topic |
Cantor sets Diophantine approximation Effective Hausdorff dimension |
| spellingShingle |
Cantor sets Diophantine approximation Effective Hausdorff dimension Becher, V. Reimann, J. Slaman, T.A. Irrationality exponent, Hausdorff dimension and effectivization |
| topic_facet |
Cantor sets Diophantine approximation Effective Hausdorff dimension |
| 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. © 2017, Springer-Verlag GmbH Austria. |
| format |
JOUR |
| author |
Becher, V. Reimann, J. Slaman, T.A. |
| author_facet |
Becher, V. Reimann, J. Slaman, T.A. |
| author_sort |
Becher, V. |
| title |
Irrationality exponent, Hausdorff dimension and effectivization |
| 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 |
| url |
http://hdl.handle.net/20.500.12110/paper_00269255_v185_n2_p167_Becher |
| work_keys_str_mv |
AT becherv irrationalityexponenthausdorffdimensionandeffectivization AT reimannj irrationalityexponenthausdorffdimensionandeffectivization AT slamanta irrationalityexponenthausdorffdimensionandeffectivization |
| _version_ |
1807315261689167872 |