A note on Hausdorff measures of quasi-self-similar sets

Computability of the Hausdorff and packing measures on self-similar sets and the self-similar tiling principle

We state a self-similar tiling principle which shows that any open subset of a self-similar set with open set condition may be tiled without loss of measure by copies under similitudes of any closed subset with positive measure. We use this method to get the optimal coverings and packings which give the exact value of the Hausdorff-type and packing measures. In particular, we show that the exac...

Hausdorff Dimension of Self-similar Sets with Overlaps

A Note on Measures of Parallel Sets

The r-parallel set to a set A in a Euclidean space consists of all points with distance at most r from A. We clarify the relation between the volume and the surface area of parallel sets and study the asymptotic behaviour of both quantities as r tends to 0. We show, for instance, that in general, the existence of a (suitably rescaled) limit of the surface area implies the existence of the corre...

Renewal of Singularity Sets of Statistically Self-similar Measures

This paper investigates new properties concerning the multifractal structure of a class of statistically self-similar measures. These measures include the well-known Mandelbrot multiplicative cascades, sometimes called independent random cascades. We evaluate the scale at which the multifractal structure of these measures becomes discernible. The value of this scale is obtained through what we ...

Self-similar Measures and Intersections of Cantor Sets

It is natural to expect that the arithmetic sum of two Cantor sets should have positive Lebesgue measure if the sum of their dimensions exceeds 1, but there are many known counterexamples, e.g. when both sets are the middle-α Cantor set and α ∈ ( 1 3 , 1 2 ). We show that for any compact set K and for a.e. α ∈ (0, 1), the arithmetic sum of K and the middle-α Cantor set does indeed have positive...