International Journal of Mathematics and Mathematical Sciences
Volume 11 (1988), Issue 4, Pages 643-649
doi:10.1155/S016117128800078X
A note on equivalent interval covering systems for Hausdorff dimension on ℝ
Department of Statistics and Actuarial Science, University of Waterloo, Ontario, Waterloo N2L 3G1, Canada
Received 30 September 1987
Copyright © 1988 C. D. Cutler. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
The Hausdorff dimension of a set in ℝ is usually defined by considering countable coverings of the set by general intervals. In this note we establish sufficient conditions under which coverings whose members are restricted to a particular family g of intervals will produce the same value for dimension. A result of Billingsley is then employed to obtain a general technique for computing the dimensions of sets defined by certain types of generalized expansions. A specific example is included.