Effective Continued Fraction Dimension versus Effective Hausdorff Dimension of Reals

TL;DR

This paper explores the relationship between effective continued fraction dimension and effective Hausdorff dimension of real numbers, revealing they can differ significantly and establishing new bounds and characterizations.

Contribution

It provides an equivalent characterization of continued fraction dimension via Kolmogorov complexity and constructs an optimal lower semi-computable s-gale for continued fractions.

Findings

Effective continued fraction and Hausdorff dimensions can differ for the same real.

Constructed an optimal lower semi-computable s-gale for continued fractions.

Proved bounds on the Lebesgue measure of continued fraction cylinders.

Abstract

We establish that constructive continued fraction dimension originally defined using $s$-gales is robust, but surprisingly, that the effective continued fraction dimension and effective (base-$b$) Hausdorff dimension of the same real can be unequal in general. We initially provide an equivalent characterization of continued fraction dimension using Kolmogorov complexity. In the process, we construct an optimal lower semi-computable $s$-gale for continued fractions. We also prove new bounds on the Lebesgue measure of continued fraction cylinders, which may be of independent interest. We apply these bounds to reveal an unexpected behavior of continued fraction dimension. It is known that feasible dimension is invariant with respect to base conversion. We also know that Martin-L\"of randomness and computable randomness are invariant not only with respect to base conversion, but also with respect to the continued fraction representation. In contrast, for any $0 < \varepsilon < 0.5$, we prove the existence of a real whose effective Hausdorff dimension is less than $\varepsilon$, but whose effective continued fraction dimension is greater than or equal to $0.5$. This phenomenon is related to the ``non-faithfulness'' of certain families of covers, investigated by Peres and Torbin and by Albeverio, Ivanenko, Lebid and Torbin. We also establish that for any real, the constructive Hausdorff dimension is at most its effective continued fraction dimension.