Recently, Stull [Stull, 2025], [Stull, 2022] resolved a long-standing open problem posed by Lutz, on whether the set of effective Hausdorff dimensions of points on a straight line in ℝ² - the effective dimension spectrum of the line - contains a unit interval. This question is related to problems in classical fractal geometry like the Kakeya conjecture and Furstenberg sets. Stull posed an open question on the dimension spectra of polynomial curves. For the first result, with new techniques which adapt the theory of classical real root-finding of polynomials to the current setting, we show that the dimension spectra of every polynomial curve contains at least two points. This answers an open question posed by Stull [Stull, 2025], [Stull, 2022]. We use the main result to construct a class of polynomials which have width strictly greater than 1, answering a second problem stated in [Stull, 2025], [Stull, 2022]. Stull [Stull, 2025] resolved the dimension spectrum conjecture for planar lines, showing that it contains a unit interval. For the second result, we resolve the conjecture for a subfamily of polynomials whose coefficients form a "low" dimension point in ℝ^{d+1}.

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.

The classical Banach-Mazur game characterizes sets of first category in a topological space. In this work, we show that an effectivized version of the game yields a characterization of sets of effective first category. Using this, we provide a game-theoretic proof of an effective theorem in dynamical systems, namely the category version of Poincaré Recurrence. The Poincaré Recurrence Theorem for category states that for a homeomorphism without open wandering sets, the set of non recurrent points forms a first category (meager) set. As an application of the effectivization of the Banach-Mazur game, we show that such a result holds true in effective settings as well.

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.

Finite-state dimension, introduced early in this century as a finite-state version of classical Hausdorff dimension, is a quantitative measure of the lower asymptotic density of information in an infinite sequence over a finite alphabet, as perceived by finite automata. Finite-state dimension is a robust concept that now has equivalent formulations in terms of finite-state gambling, lossless finite-state data compression, finite-state prediction, entropy rates, and automatic Kolmogorov complexity. The 1972 Schnorr-Stimm dichotomy theorem gave the first automata-theoretic characterization of normal sequences, which had been studied in analytic number theory since Borel defined them in 1909. This theorem implies, in present-day terminology, that a sequence (or a real number having this sequence as its base-b expansion) is normal if and only if it has finite-state dimension 1. One of the most powerful classical tools for investigating normal numbers is the 1916 Weyl’s criterion, which characterizes normality in terms of exponential sums. Such sums are well studied objects with many connections to other aspects of analytic number theory, and this has made use of Weyl’s criterion especially fruitful. This raises the question whether Weyl’s criterion can be generalized from finite-state dimension 1 to arbitrary finite-state dimensions, thereby making it a quantitative tool for studying data compression, prediction, etc. i.e., Can we characterize all compression ratios using exponential sums?. This paper does exactly this. We extend Weyl’s criterion from a characterization of sequences with finite-state dimension 1 to a criterion that characterizes every finite-state dimension. This turns out not to be a routine generalization of the original Weyl criterion. Even though exponential sums may diverge for non-normal numbers, finite-state dimension can be characterized in terms of the dimensions of the subsequence limits of the exponential sums. In case the exponential sums are convergent, they converge to the Fourier coefficients of a probability measure whose dimension is precisely the finite-state dimension of the sequence. This new and surprising connection helps us bring Fourier analytic techniques to bear in proofs in finite-state dimension, yielding a new perspective. We demonstrate the utility of our criterion by substantially improving known results about preservation of finite-state dimension under arithmetic. We strictly generalize the results by Aistleitner and Doty, Lutz and Nandakumar for finite-state dimensions under arithmetic operations. We use the method of exponential sums and our Weyl criterion to obtain the following new result: If y is a number having finite-state strong dimension 0, then dim_FS(x+qy) = dim_FS(x) and Dim_FS(x+qy) = Dim_FS(x) for any x ∈ ℝ and q ∈ ℚ. This generalization uses recent estimates obtained in the work of Hochman [Hochman, 2014] regarding the entropy of convolutions of probability measures.

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.

This work solves an open question in finite-state compressibility posed by Lutz and Mayordomo about compressibility of real numbers in different bases. Finite-state compressibility, or equivalently, finite-state dimension, quantifies the asymptotic lower density of information in an infinite sequence. Absolutely normal numbers, being finite-state incompressible in every base of expansion, are precisely those numbers which have finite-state dimension equal to $1$ in every base. At the other extreme, for example, every rational number has finite-state dimension equal to $0$ in every base. Generalizing this, Lutz and Mayordomo (2021) posed the question: are there numbers which have absolute positive finite-state dimension strictly between 0 and 1 - equivalently, is there a real number $ξ$ and a compressibility ratio $s \in (0,1)$ such that for every base $b$, the compressibility ratio of the base-$b$ expansion of $ξ$ is precisely $s$? It is conceivable that there is no such number. Indeed, some works explore ``zero-one'' laws for other feasible dimensions - i.e. sequences with certain properties either have feasible dimension 0 or 1, taking no value strictly in between. However, we answer the question of Lutz and Mayordomo affirmatively by proving a more general result. We show that given any sequence of rational numbers $\langle q_b \rangle$, we can explicitly construct a single number $ξ$ such that for any base $b$, the finite-state dimension/compression ratio of $ξ$ in base-$b$ is $q_b$. As a special case, this result implies the existence of absolutely dimensioned numbers for any given rational dimension between $0$ and $1$, as posed by Lutz and Mayordomo. In our construction, we combine ideas from Wolfgang Schmidt's construction of absolutely normal numbers (1962), results regarding low discrepancy sequences and several new estimates related to exponential sums.

We initiate the study of effective pointwise ergodic theorems in resource-bounded settings. Classically, the convergence of the ergodic averages for integrable functions can be arbitrarily slow [Ulrich Krengel, 1978]. In contrast, we show that for a class of PSPACE L¹ functions, and a class of PSPACE computable measure-preserving ergodic transformations, the ergodic average exists and is equal to the space average on every EXP random. We establish a partial converse that PSPACE non-randomness can be characterized as non-convergence of ergodic averages. Further, we prove that there is a class of resource-bounded randoms, viz. SUBEXP-space randoms, on which the corresponding ergodic theorem has an exact converse - a point x is SUBEXP-space random if and only if the corresponding effective ergodic theorem holds for x.

Recently, Scheerer [Adrian-Maria Scheerer, 2017] and Vandehey [Vandehey, 2016] showed that normality for continued fraction expansions and base-b expansions are incomparable notions. This shows that at some level, randomness for continued fractions and binary expansion are different statistical concepts. In contrast, we show that the continued fraction expansion of a real is computably random if and only if its binary expansion is computably random. To quantify the degree to which a continued fraction fails to be effectively random, we define the effective Hausdorff dimension of individual continued fractions, explicitly constructing continued fractions with dimension 0 and 1.

Abstract Jockusch showed that 2-generic degrees are downward dense below a 2-generic degree. That is, if a is 2-generic, and $0 < {\bf{b}} < {\bf{a}}$ , then there is a 2-generic g with $0 < {\bf{g}} < {\bf{b}}.$ In the case of 1-generic degrees Kumabe, and independently Chong and Downey, constructed a minimal degree computable from a 1-generic degree. We explore the tightness of these results. We solve a question of Barmpalias and Lewis-Pye by constructing a minimal degree computable from a weakly 2-generic one. While there have been full approximation constructions of ${\rm{\Delta }}_3^0$ minimal degrees before, our proof is rather novel since it is a computable full approximation construction where both the generic and the minimal degrees are ${\rm{\Delta }}_3^0 - {\rm{\Delta }}_2^0$ .

In this paper we propose a quantification of distributions on a set of strings, in terms of how close to pseudorandom a distribution is. The quantification is an adaptation of the theory of dimension of sets of infinite sequences introduced by Lutz. Adapting Hitchcock's work, we also show that the logarithmic loss incurred by a predictor on a distribution is quantitatively equivalent to the notion of dimension we define. Roughly, this captures the equivalence between pseudorandomness defined via indistinguishability and via unpredictability. Later we show some natural properties of our notion of dimension. We also do a comparative study among our proposed notion of dimension and two well known notions of computational analogue of entropy, namely HILL-type pseudo min-entropy and next-bit pseudo Shannon entropy. Further, we apply our quantification to the following problem. If we know that the dimension of a distribution on the set of n-length strings is s in (0,1], can we extract out O(sn) pseudorandom bits out of the distribution? We show that to construct such extractor, one need at least Omega(log n) bits of pure randomness. However, it is still open to do the same using O(log n) random bits. We show that deterministic extraction is possible in a special case - analogous to the bit-fixing sources introduced by Chor et al., which we term nonpseudorandom bit-fixing source. We adapt the techniques of Gabizon, Raz and Shaltiel to construct a deterministic pseudorandom extractor for this source. By the end, we make a little progress towards P vs. BPP problem by showing that existence of optimal stretching function that stretches O(log n) input bits to produce n output bits such that output distribution has dimension s in (0,1], implies P=BPP.

This work is a synthesis of recent advances in computable analysis with the theory of algorithmic randomness. In this theory, we try to strengthen probabilistic laws, i.e., laws which hold with probability 1, to laws which hold in their pointwise effective form - i.e., laws which hold for every individual constructively random point. In a tour-de-force, V'yugin proved an effective version of the Ergodic Theorem which holds when the probability space, the transformation and the random variable are computable. However, V'yugin's Theorem cannot be directly applied to many examples, because all computable functions are continuous, and many applications use discontinuous functions.