An introduction to Kolmogorov complexity and its applications (2nd ed.)
An introduction to Kolmogorov complexity and its applications (2nd ed.)
A Kolmogorov complexity characterization of constructive Hausdorff dimension
Information Processing Letters
Random elements in effective topological spaces with measure
Information and Computation
The dimensions of individual strings and sequences
Information and Computation
Correspondence Principles for Effective Dimensions
Theory of Computing Systems
FOCS '06 Proceedings of the 47th Annual IEEE Symposium on Foundations of Computer Science
The arithmetical complexity of dimension and randomness
ACM Transactions on Computational Logic (TOCL)
Computability of probability measures and Martin-Löf randomness over metric spaces
Information and Computation
| Hi-index | 0.01 |
This paper initiates the study of sets in Euclidean space R^n(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^I consisting of all points in R^n 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.