Conditional Dimension in Metric Spaces: A Natural Metric-Space Counterpart of Kolmogorov-Complexity-Based Mutual Dimension

It is known that dimension of a set in a metric space can be characterized in information-related terms – in particular, in terms of Kolmogorov complexity of different points from this set. The notion of Kolmogorov complexity K(x) – the shortest length of a program that generates a sequence x – can be naturally generalized to conditional Kolmogorov complexity K(x : y) – the shortest length of a program that generates x by using y as an input. It is therefore reasonable to use conditional Kolmogorov complexity to formulate a conditional analogue of dimension. Such a generalization has indeed been proposed, under the name of mutual dimension. However, somewhat surprisingly, this notion was formulated in pure Kolmogorov-complexity terms, without any analysis of possible metric-space meaning. In this paper, we describe the corresponding metric-space notion of conditional dimension – a natural metric-space counterpart of the Kolmogorov-complexity-based mutual dimension. 1 Need for a Metric Analogue of Mutual Dimension: Formulation of a Problem What is dimension: an informal idea. A straight line segment S1 is a 1-dimensional set, meaning that to select a point on this segment, it is sufficient to describe the value of a single real-valued quantity. Similarly, a planar area S2 is a 2-dimensional set meaning that to select a point in this area, we need to describe the values of two real-valued quantities: namely, two coordinates of this point. A spatial area S3 is a 3-dimensional set meaning that to select a point in