Constructive Dimension and Hausdorff Dimension: The Case of Exact Dimension

The present paper generalises results by Lutz and Ryabko. We prove a martingale characterisation of exact Hausdorff dimension. On this base we introduce the notion of exact constructive dimension of (sets of) infinite strings. Furthermore, we generalise Ryabko’s result on the Hausdorff dimension of the set of strings having asymptotic Kolmogorov complexity ≤ α to the case of exact dimension. The papers [13,14,16,17,7,8] show a close connection between Hausdorff dimension and constructive dimension or, equivalently, asymptotic Kolmogorov complexity of (sets of) infinite strings. In all these papers,the Hausdorff dimension of a set is defined as usual (cf. [3,4]) to be a real number. It is interesting to observe that already Hausdorff in his paper [6] defined the (fractal) dimension of a set to be a real function of a special shape. To distinguish it from the “usual” Hausdorff dimension Hausdorff’s original definition is referred to as exact Hausdorff dimension [9,5,10]. The aim of the present paper is to generalise results by Lutz [7,8] and Ryabko [13] to the case of this exact dimension. First we deal with the martingale characterisation of Hausdorff dimension [7,8]. This leads in a natural way to a definition of exact constructive dimension. From this we derive the particularly interesting fact that the exact dimension of an infinite string ξ can be identified with Levin’s [22] universal left computable continuous semi-measure restricted to the set of finite prefixes of ξ. As a further consequence we obtain a connection to the a priori complexity (cf. [20,21]) of finite strings yielding just another proof that constructive dimension equals Kolmogorov complexity (cf. [18]). Having a priori complexity in mind we generalise Ryabko’s result that the set of infinite strings having asymptotic Kolmogorov complexity ≤ α has Hausdorff dimension α to the case of exact dimensions. Finally we apply our results to the family of functions of the logarithmic scale, as considered by Hausdorff [6]. Here we show that, unlike the case of asymptotic ISSN 1433-8092 Electronic Colloquium on Computational Complexity, Report No. 74 (2011) Electronic Colloquium on Computational C mplexity, Revision 1 f Report No. 74 (2011)