Staiger, Ludwig Combinatorial properties of the Hausdorff dimension. (English) Zbl 0709.11041 J. Stat. Plann. Inference 23, No. 1, 95-100 (1989). The author considers subsets F of \(X^{\omega}\), the space of infinite sequences over an alphabet of cardinality r. Introducing a suitable entropy \(H_ F\) for such sets (in terms of combinatorial properties) he shows that, for a certain class of such sets, \(H_ F\) equals the Hausdorff dimension of the set obtained by interpreting each \(x\in F\) as the base r expansion of a real number. Reviewer: B.Volkmann Cited in 6 Documents MSC: 11K55 Metric theory of other algorithms and expansions; measure and Hausdorff dimension 68Q45 Formal languages and automata 94A17 Measures of information, entropy Keywords:channel capacity; structure function; alphabet; entropy; Hausdorff dimension PDF BibTeX XML Cite \textit{L. Staiger}, J. Stat. Plann. Inference 23, No. 1, 95--100 (1989; Zbl 0709.11041) Full Text: DOI References: [1] Billingsley, P., Ergodic Theory and Information (1965), Wiley: Wiley New York · Zbl 0141.16702 [2] Cajar, H., Billingsley Dimension in Probability Spaces, (Lect. Notes Math. No. 892 (1981), Springer-Verlag: Springer-Verlag Berlin) · Zbl 0508.60002 [3] Conner, W. M., The capacity and ambiguity of a transducer, Ann. Math. Statist., 41, 2093-2104 (1970) · Zbl 0215.31001 [4] Kuich, W., On the entropy of context-free languages, Inform. and Control, 16, 173-200 (1970) · Zbl 0193.32603 [5] Shannon, C. E., A mathematical theory of communication, Bell System Tech. J., 27, 623-656 (1948) · Zbl 1154.94303 [6] Staiger, L., On the relative density of sources, (Trans. 9th Prague Conference, Vol. B (1983), Academia: Academia Prague), 185-188 [7] Staiger, L., The entropy of finite-state \(ω\)-languages, Problems Control Inform. Theory, 14, 383-392 (1985) · Zbl 0582.94012 [8] Volkmann, B., Über Hausdorffsche Dimensionen von Mengen, die durch Zifferneigenschaften charakterisiert sind I, III and V, Math. Zeitschr., 65, 389-413 (1956) · Zbl 0071.05303 [9] Cameron, P. J., Portrait of a typical sum-free set, (Whitehead, C., Surveys in Combinatorics 1987 (1987), Cambridge Univ. Press: Cambridge Univ. Press Cambridge), 13-42, London Math. Soc. Lect. Notes No. 123 · Zbl 0677.05063 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.