Monatshefte für Mathematik

, Volume 185, Issue 2, pp 167–188

# Irrationality exponent, Hausdorff dimension and effectivization

• Verónica Becher
• Jan Reimann
• Theodore A. Slaman
Article

## 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.

### Keywords

Diophantine approximation Cantor sets Effective Hausdorff dimension

### Mathematics Subject Classification

11J82 11J83 03D32

## Notes

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## Authors and Affiliations

• Verónica Becher
• 1
• 2
• Jan Reimann
• 3
• Theodore A. Slaman
• 4
1. 1.Departamento de Computación, Facultad de Ciencias Exactas y NaturalesUniversidad de Buenos AiresBuenos AiresArgentina
2. 2.CONICETBuenos AiresArgentina
3. 3.Department of MathematicsPenn State UniversityUniversity ParkUSA
4. 4.Department of MathematicsUniversity of California, BerkeleyBerkeleyUSA