Homogeneity and non-coincidence of Hausdorff and box dimensions for subsets of $\mathbb{R}^n$
Volume 181 / 2007
Studia Mathematica 181 (2007), 285-296
MSC: 28A78, 28A80.
DOI: 10.4064/sm181-3-5
Abstract
A class of subsets of $\mathbb{R}^n$ is constructed that have certain homogeneity and non-coincidence properties with respect to Hausdorff and box dimensions. For each triple $(r,s,t)$ of numbers in the interval $(0,n]$ with $r< s< t$, a compact set $K$ is constructed so that for any non-empty subset $U$ relatively open in $K$, we have $(\dim_{\rm H}(U), \underline{\dim}_{\rm B}(U), \overline{\dim}_{\rm B}(U))=(r, s, t)$. Moreover, $2^{-n}\leq H^{r}(K)\leq 2n^{{r}/{2}}$.
