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• W2010424305 abstract "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)" @default.
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• W2010424305 date "2008-08-27" @default.
• W2010424305 modified "2023-09-25" @default.
• W2010424305 title "Connectivity properties of dimension level sets" @default.
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• W2010424305 doi "https://doi.org/10.1002/malq.200710060" @default.
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