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 non-negative real 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.

Files

Metadata

Work Title Irrationality Exponent, Hausdorff Dimension and Effectivization
Access
Open Access
Creators
  1. Reimann, Jan Severin
  2. Becher, Veronica
  3. Slaman, Theodore A.
Keyword
  1. Hausdorff Dimension
  2. Number Theory
  3. Diophantine Approximation
  4. Kolmogorov Complexity
License Attribution-NonCommercial-NoDerivs 3.0 United States
Work Type Article
Deposited January 09, 2016

Versions

Analytics

Collections

This resource is currently not in any collection.

Work History

Version 1
published

  • Created
  • Added 4mw22v461s_version1_eie.pdf
  • Added Creator Reimann, Jan Severin
  • Added Creator Becher, Veronica
  • Added Creator Slaman, Theodore A.
  • Published