Homogeneity and non-coincidence of Hausdorff and box dimensions for subsets of $\mathbb{R}^n$

Anders Nilsson ¹   ; Peter Wingren ¹

¹ Department of Mathematics and Mathematical Statistics Umeå University 901 87 Umeå, Sweden

Studia Mathematica, Tome 181 (2007) no. 3, pp. 285-296

Résumé

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, $2^{-n}\leq H^{r}(K)\leq 2n^{{r}/{2}}$.

Détail

DOI : 10.4064/sm181-3-5

Keywords: class subsets mathbb constructed have certain homogeneity non coincidence properties respect hausdorff box dimensions each triple numbers interval compact set constructed non empty subset relatively have dim underline dim overline dim moreover n leq leq