Connectivity Properties of Dimension Level Sets

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