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Connectivity properties of dimension level sets

This paper initiates the study of sets in Euclidean spaces ℝn (n ≥ 2) that are defined in terms of the dimensions of their elements. Specifically, given an interval I ⊆ [0, n ], we are interested in the connectivity properties of the set DIMI , consisting of all points in ℝn whose (constructive Hausdorff) dimensions lie in I, and of its dual DIMI str, consisting of all points whose strong (constructive packing) dimensions lie in I. If I is [0, 1) or (n – 1, n ], it is easy to see that the sets DIMI and DIMI str are totally disconnected. In contrast, we show that if I is [0, 1] or [n – 1, n ], then the sets DIMI and DIMI str are path‐connected. Our proof of this fact uses geometric properties of Kolmogorov complexity in Euclidean spaces. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)