A constructive version of Hausdorff dimension is developed using constructive supergales, which are betting strategies that generalize the constructive supermartingales used in the theory of individual random sequences. This constructive dimension is used to assign every individual (infinite, binary) sequence S a dimension, which is a real number dim(S) in the interval [0, 1]. Sequences that

We discuss selected topics of current research interest in the theory of dynamical systems, with emphasis on dimension theory, multifractal analysis, and quantitative recurrence. The topics include the quantitative versus the qualitative behavior of Poincare recurrence, the product structure of invariant measures and return times, the dimension of invariant sets and invariant measures, the complexity of the level sets of local quantities from the point of view of Hausdorff dimension, and the conditional variational principles as well as their applications to problems in number theory.

ABSTRACT. We continue the study of effective Hausdorff dimension as it was initiated by LUTZ. Whereas he uses a generalization of martingales on the Cantor space to introduce this notion we give a characterization in terms of effective s-dimensional Hausdorff measures, similar to the effectivization of Lebesgue measure by MARTIN-LÖF. It turns out that effective Hausdorff dimension allows to classify sequences according to their ‘degree ’ of algorithmic randomness, i.e., their algorithmic density of information. Earlier the works of STAIGER and RYABKO showed a deep connection between Kolmogorov complexity and Hausdorff dimension. We further develop this relationship and use it to give effective versions of some important properties of (classical) Hausdorff dimension. Finally, we determine the effective dimension of some objects arising in the context of computability theory, such as degrees and spans. 1.

We study the Hausdorff dimension of a large class of sets in the real line defined in terms of the distribution of frequencies of digits for the representation in some integer base. In particular, our results unify and extend classical work of Borel, Besicovitch, Eggleston, and Billingsley in several directions. Our methods are based on recent results concerning the multifractal analysis of dynamical systems and often allow us to obtain explicit expressions for the Hausdorff dimension.

digit frequencies constraints

Abstract. We investigate the size and large intersection properties of Et = {x ∈ R d | ‖x − k − xi ‖ < ri t for infinitely many (i, k) ∈ I µ,α × Z d}, where d ∈ N, t ≥ 1, I is a denumerable set, (xi, ri)i∈I is a family in [0,1] d × (0, ∞) and I µ,α denotes the set of all i ∈ I such that the µ-mass of the ball with center xi and radius ri behaves as ri α for a given Borel measure µ and a given α> 0. We establish that the set Et belongs to the class G h (R d) of sets with large intersection with respect to a certain gauge function h, provided that (xi, ri)i∈I is a heterogeneous ubiquitous system with respect to µ. In particular, Et has infinite Hausdorff g-measure for every gauge function g that increases faster than h in a neighborhood of zero. We also give several applications to metric number theory. 1.

Abstract. We prove that systems satisfying the specification property are saturated in the sense that the topological entropy of the set of generic points of any invariant measure is equal to the measure-theoretic entropy of the measure. We study Banach valued Birkhoff ergodic averages and obtain a variational principle for its topological entropy spectrum. As application, we examine a particular example concerning with the set of real numbers for which the frequencies of occurrences in their dyadic expansions of infinitely many words are prescribed. This relies on our explicit determination of a maximal entropy measure. 1.

Properties of the set Ts of ”particularly non-normal numbers ” of the unit interval are studied in details (Ts consists of real numbers x, some of whose s-adic digits have the asymptotic frequencies in the nonterminating s − adic expansion of x, and some do not). It is proven that the set Ts is residual in the topological sense (i.e., it is of the first Baire category) and it is generic in the sense of fractal geometry (Ts is a superfractal set, i.e., its Hausdorff-Besicovitch dimension is equal to 1). A topological and fractal classification of sets of real numbers via analysis of asymptotic frequencies of digits in their s-adic expansions is presented.

Abstract. This paper is mainly concerned with Hausdorff dimensions of Besicovitch-Eggleston subsets in countable symbolic space. A notable point is that, the dimension values posses a universal lower bound depending only on the underlying metric. As a consequence of the main results, we obtain Hausdorff dimension formulas for sets of real numbers with prescribed digit frequencies in their Lüroth expansions.

We transfer classical results on the Hausdorff dimension of b-adic and continued fraction expansions of real numbers to another expansion.