Self-similar sets with optimal coverings and packings

Israel Journal of Mathematics, 2000

1999

We develop a new combinatorial method to estimate Hausdorff measures of various self-similar sets. This method can be applied to the evaluation of Hausdorff measures which are induced by various Hausdorff functions including power functions. Moreover, a few examples for evaluations of the lower and upper bounds of Hausdorff measures of uniform Cantor sets are introduced.

Chaos, Solitons & Fractals, 2012

Topological Methods in Nonlinear Analysis, 2015

Proceedings of the American Mathematical Society, 1992

Real Analysis Exchange, 2002

Advances in Mathematics, 2007

2022

In this paper we answer a question raised by David H. Fremlin about the Hausdorff measure of R 2 with respect to a distance inducing the Euclidean topology. In particular we prove that the Hausdorff n -dimensional measure of R n is never 0 when considering a distance inducing the Euclidean topology. Finally, we show via counterexamples that the previous result does not hold in general if we remove the assumption on the topology.

Proceedings of the American Mathematical Society, 1992