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.

翻译：我们证明，最初利用$s$-赌博机定义的结构性连分数维数是稳健的，但令人惊讶的是，同一实数的有效连分数维数和有效（基-$b$）豪斯多夫维数通常可能不相等。我们首先利用柯尔莫哥洛夫复杂度为连分数维数提供了等价刻画。在此过程中，我们构造了一个针对连分数的最优下半可计算$s$-赌博机。我们还证明了连分数圆柱体的勒贝格测度的新界，这可能具有独立意义。我们应用这些界揭示了连分数维数的一个意外行为。已知可行维数在基变换下具有不变性。我们也知道马丁-洛夫随机性和可计算随机性不仅对基变换不变，而且对连分数表示也不变。相反，对于任意$0 < \varepsilon < 0.5$，我们证明了存在一个实数，其有效豪斯多夫维数小于$\varepsilon$，但其有效连分数维数大于或等于$0.5$。这一现象与佩雷斯和托尔宾以及阿尔贝韦里奥、伊万年科、莱比德和托尔宾所研究的某些覆盖族的“非忠实性”有关。我们还证明，对于任意实数，结构性豪斯多夫维数至多为其有效连分数维数。