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 t...

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Bibliographic Details
Published in:Monatshefte für Mathematik 2018-02, Vol.185 (2), p.167-188
Main Authors: Becher, Verónica, Reimann, Jan, Slaman, Theodore A.
Format: Article
Language:English
Subjects:
Irrationality
Mathematics
Mathematics and Statistics
Real numbers
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Summary: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.
ISSN:0026-9255
1436-5081
DOI:10.1007/s00605-017-1094-2