Irrationality Exponent, Hausdorff Dimension and Effectivization
We generalize the classical theorem by Jarník and Besicovitch on the irrationality exponents of real numbers and Hausdorff dimension. Let a be any real number greater than or equal to 2 and let b be any nonnegative real less than or equal to 2/a. We show that there is a Cantorlike 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 Cantorlike 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.
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Work Title  Irrationality Exponent, Hausdorff Dimension and Effectivization 

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License  AttributionNonCommercialNoDerivs 3.0 United States 
Work Type  Article 
Deposited  January 09, 2016 
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