Characterization of local dimension functions of subsets of ${\Bbb R}^{d}$

L. Olsen¹

¹ Department of Mathematics University of St. Andrews St. Andrews, Fife KY16 9SS, Scotland

Colloquium Mathematicum, Tome 103 (2005) no. 2, pp. 231-239

Résumé

For a subset $E\subseteq {\mathbb R}^{d}$ and $x\in {\mathbb R}^{d}$, the local Hausdorff dimension function of $E$ at $x$ is defined by $$ \mathop {\rm dim} \nolimits _{{\mathsf H}, {\mathsf loc}}(x,E) = \mathop {\rm lim}_{r\searrow 0}\mathop {\rm dim}\nolimits _{ {\sf H}}(E\cap B(x,r)) $$ where $\mathop {\rm dim}\nolimits _{{\sf H}}$ denotes the Hausdorff dimension. We give a complete characterization of the set of functions that are local Hausdorff dimension functions. In fact, we prove a significantly more general result, namely, we give a complete characterization of those functions that are local dimension functions of an arbitrary regular dimension index.

Détail

DOI : 10.4064/cm103-2-8

Keywords: subset subseteq mathbb mathbb local hausdorff dimension function defined mathop dim nolimits mathsf mathsf loc mathop lim searrow mathop dim nolimits cap where mathop dim nolimits denotes hausdorff dimension complete characterization set functions local hausdorff dimension functions prove significantly general result namely complete characterization those functions local dimension functions arbitrary regular dimension index