A note on equivalent interval covering systems for Hausdorff dimension onℝ

Abstract

The Hausdorff dimension of a set inℝis 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 familygof 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.