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Abstract:

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.

Registro:

Documento: Artículo
Título:Irrationality exponent, Hausdorff dimension and effectivization
Autor:Becher, V.; Reimann, J.; Slaman, T.A.
Filiación:Departamento de Computación, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, Pabellón I, Ciudad Universitaria, Buenos Aires, 1428, Argentina
CONICET, Buenos Aires, Argentina
Department of Mathematics, Penn State University, 318B McAllister, University Park, PA 16802, United States
Department of Mathematics, University of California, Berkeley, 719 Evans Hall #3840, Berkeley, CA 94720-3840, United States
Palabras clave:Cantor sets; Diophantine approximation; Effective Hausdorff dimension
Año:2018
Volumen:185
Número:2
Página de inicio:167
Página de fin:188
DOI: http://dx.doi.org/10.1007/s00605-017-1094-2
Título revista:Monatshefte fur Mathematik
Título revista abreviado:Monatsh. Math.
ISSN:00269255
Registro:https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_00269255_v185_n2_p167_Becher

Referencias:

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Citas:

---------- APA ----------
Becher, V., Reimann, J. & Slaman, T.A. (2018) . Irrationality exponent, Hausdorff dimension and effectivization. Monatshefte fur Mathematik, 185(2), 167-188.
http://dx.doi.org/10.1007/s00605-017-1094-2
---------- CHICAGO ----------
Becher, V., Reimann, J., Slaman, T.A. "Irrationality exponent, Hausdorff dimension and effectivization" . Monatshefte fur Mathematik 185, no. 2 (2018) : 167-188.
http://dx.doi.org/10.1007/s00605-017-1094-2
---------- MLA ----------
Becher, V., Reimann, J., Slaman, T.A. "Irrationality exponent, Hausdorff dimension and effectivization" . Monatshefte fur Mathematik, vol. 185, no. 2, 2018, pp. 167-188.
http://dx.doi.org/10.1007/s00605-017-1094-2
---------- VANCOUVER ----------
Becher, V., Reimann, J., Slaman, T.A. Irrationality exponent, Hausdorff dimension and effectivization. Monatsh. Math. 2018;185(2):167-188.
http://dx.doi.org/10.1007/s00605-017-1094-2