Finite-state dimension, introduced as a finite-state analogue of Hausdorff dimension, quantifies the lower asymptotic density of information in an infinite sequence as perceived by finite-state automata. It admits several equivalent formulations; two particularly useful are via finite-state gambling strategies and via the optimal asymptotic compression ratio achieved by information-lossless finite-state compressors. Normal sequences represent the highest level of algorithmic randomness visible to finite automata, and are exactly those sequences having finite-state dimension equal to 1. This motivates a bounded-memory notion of randomness extraction: can a finite-state transducer, reading a single sequence streamingly, extract a normal output from a single input source? More modestly, can it always transform the input into an output of strictly higher finite-state dimension? Finite-state transducers can perform surprisingly effective one-pass transformations: even with constant memory they can implement variable-length coding schemes including Shannon-Fano coding, remove local redundancy, and increase the apparent randomness rate on many structured or stochastic inputs. We show randomness extraction using transducers is impossible in a strong, explicit form. For every rational s ∈ (0,1), we construct a near linear-time computable binary sequence X with dim_FS(X) = s such that for every finite-state transducer T, the output satisfies dim_FS(T(X)) ≤ s. Thus, for these sequences, finite-state transduction cannot extract normality - indeed it cannot even improve finite-state dimension. Our proof proceeds by a structural analysis of finite-state transducers together with a dimension-preserving diagonal construction that, for each target s, builds a sequence whose organization defeats every such transducer’s attempt to concentrate randomness. The result is a finite-state analogue of Miller’s non-extractability phenomenon for effective dimension, but its proof relies on substantially different techniques, tailored to the finite-state setting. Furthermore, we show that the impossibility persists even with multiple independent input streams. We treat two notions of independence: (i) Kolmogorov-complexity–based independence (via joint prefix complexity), and (ii) a finite-state notion of relative independence, formulated via relative finite-state dimension. By sharp contrast with the effective-dimension setting - where two independent sources suffice for a uniform effective procedure that boosts randomness rate arbitrarily close to 1 - we show that finite-state dimension exhibits no comparable multi-source extraction phenomenon. Specifically, for every rational s ∈ (0,1) and every fixed k ≥ 2, there exist k independent sources, each of finite-state dimension s, such that for every k-input finite-state transducer T, the output satisfies dim_FS(T(X_1,… ,X_k)) ≤ s. Thus, even independent streams do not allow bounded-memory transduction to output a normal sequence or to increase finite-state dimension.

Finite-state dimension quantifies the asymptotic density of information in an infinite sequence as seen by finite automata, and can be viewed as a bounded-memory analogue of Hausdorff dimension. The theory is by now well developed for base-b expansions. In this paper we initiate the study of finite-state dimension in the setting of continued fractions. This setting brings several new difficulties. The natural reference measure is the Gauss measure, which is the canonical invariant probability measure for the Gauss transformation governing the continued fraction shift. Unlike in the base-b setting, however, this measure is not a product measure. In addition, the digit space is inherently infinite, so the symbolic setting is markedly less rigid than the usual finite-alphabet framework. Continued fraction analogues of effective dimension have received considerable attention in recent years, but a finite-state theory in this setting has so far been missing. Finite-state dimension in the classical setting admits several equivalent formulations. We begin from the original finite-state s-gale viewpoint. In the continued fraction setting, however, the betting odds are no longer fixed in advance: unlike in the base-b case, the fair payoff for the next symbol is determined by conditional Gauss probabilities, and these vary with the current continued fraction cylinder. This makes the choice of a finite-state betting model genuinely nontrivial. We first examine a natural local finite-state betting model for truncated continued fraction digits, in which a gambler may use both the visible local context and an additional finite internal memory. We show that this model is too strong: there is a fixed finite-state gambler that wins at a positive exponential rate on every continued fraction normal point. Consequently, full dimension in this model does not characterize continued fraction normality, and the classical Schnorr-Stimm dichotomy fails. We then isolate the source of this failure and introduce a restricted context-gambler model in which the gambler is allowed to use only the visible truncated context, with no additional hidden finite-state memory. For this restricted model, we obtain, in direct analogy with the base-b setting, an entropy-rate characterization of finite-state dimension in terms of conditional entropy rates, and from this derive an exact finite-state characterization of continued fraction normality: an irrational real has finite-state dimension 1 if and only if its continued fraction expansion is normal. Finally, we prove that the Schnorr-Stimm dichotomy does hold in this restricted setting: on a continued fraction normal point every such gambler either preserves constant capital or loses at an exponential rate, while on every non-normal point some such gambler wins at an exponential rate.

Hausdorff $Φ$-dimension is a notion of Hausdorff dimension developed using a restricted class of coverings of a set. We introduce an effective version of Hausdorff $Φ$-dimension, which we call constructive $Φ$-dimension. We prove a point-to-set principle for $Φ$-dimension. We also provide a characterization of constructive $Φ$-dimension using Kolmogorov complexity and $s$-gales. Finally, we apply these tools to study faithfulness of coverings $Φ$. A family of coverings $Φ$ is said to be faithful to Hausdorff dimension if the $Φ$-dimension and Hausdorff dimension coincide for every set. Similarly, $Φ$ is said to be faithful to constructive dimension if the constructive $Φ$-dimension and constructive dimension coincide for every set. We derive the necessary and sufficient conditions for the constructive dimension faithfulness of the coverings generated by the Cantor series expansion, based on the terms of the expansion. Using the point-to-set principle for Cantor coverings, we show that the same condition characterises Hausdorff dimension faithfulness of Cantor coverings, thereby giving an information theoretic proof of the result by Albeverio, Ivanenko, Lebid, and Torbin. We investigate the question of weather the notions of faithfulness at Hausdorff and constructive levels are equivalent. Using a new technique for the construction of sequences satisfying a certain Kolmogorov complexity condition, we show that the notions of ``faithfulness'' of Cantor coverings at the Hausdorff and constructive levels are equivalent, independent of the log-limit condition.

We establish that constructive continued fraction dimension originally defined using s-gales [Nandakumar and Vishnoi, 2022] 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 [Hitchcock and Mayordomo, 2013]. 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 [Nandakumar and Vishnoi, 2022]. In contrast, for any 0 < ε < 0.5, we prove the existence of a real whose effective Hausdorff dimension is less than ε, 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 [Peres and Torbin] and by Albeverio, Ivanenko, Lebid and Torbin [Albeverio et al., 2020]. We also establish that for any real, the constructive Hausdorff dimension is at most its effective continued fraction dimension.