Effective Continued Fraction Dimension versus Effective Hausdorff Dimension of Reals

We establish that effective continued fraction dimension originally defined using s -gales [ 21 ] 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. 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 effective dimension is invariant with respect to base conversion [ 8 ]. We also know that Martin-Löf randomness and computable randomness are invariant not only with respect to base conversion, but also with respect to the continued fraction representation [ 21 ]. In contrast, for any \(0 \lt \varepsilon \lt 0.5\) , 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 [ 1 , 23 ]. We also establish that, for any real number, the effective continued fraction dimension of the real number is always greater than or equal to its effective Hausdorff dimension.