Connectivity properties of dimension level sets

Abstract

This paper initiates the study of sets in Euclidean space R n (n ≥ 2) that are defined in terms of the dimensions of their elements. Specifically, given an interval I ⊆ [0, 1], we are interested in the connectivity properties of the set DIM I consisting of all points in R n whose (constructive Hausdorff) dimensions lie in the interval I. It is easy to see that the sets DIM [0,1) and DIM (n−1,n] are totally disconnected. In contrast, we show that the sets DIM [0,1] and DIM [n−1,n] are path-connected. Our proof of this fact uses geometric properties of Kolmogorov complexity in Euclidean space.

Dimensions of Points in Self-Similar Fractals

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Functions that preserve p-randomness

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