Connectivity properties of dimension level sets

✍ Scribed by Jack H. Lutz; Klaus Weihrauch

• Publisher: John Wiley and Sons
• Year: 2008
• Tongue: English
• Weight: 137 KB
• Volume: 54
• Category: Article
• ISSN: 0044-3050
• DOI: 10.1002/malq.200710060

✦ Synopsis

Abstract

This paper initiates the study of sets in Euclidean spaces ℝ^n^ (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 DIM^I^ , consisting of all points in ℝ^n^ whose (constructive Hausdorff) dimensions lie in I, and of its dual DIM^I^ ~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 DIM^I^ and DIM^I^ ~str~ are totally disconnected. In contrast, we show that if I is [0, 1] or [n – 1, n ], then the sets DIM^I^ and DIM^I^ ~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)