Irrationality Exponent, Hausdorff Dimension and Effectivization

TL;DR

This paper constructs specific Cantor-like sets to demonstrate the relationship between irrationality exponents and Hausdorff dimensions, extending classical theorems with effective and measure-theoretic insights.

Contribution

It generalizes classical results by constructing Cantor-like sets with prescribed irrationality exponents and Hausdorff dimensions, linking measure theory and effective dimension.

Findings

Existence of Cantor-like sets with prescribed Hausdorff dimension and irrationality exponent.

Almost all elements in these sets have the specified irrationality exponent.

The sets are constructed as paths in a tree of Cantor sets.

Abstract

We generalize the classical theorem by Jarnik 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.