Connectivity Properties of Dimension Level Sets

This paper initiates the study of sets in Euclidean space 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 DIMI consisting of all points in Rn 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.