Characterization of local dimension functions of subsets of ${\Bbb R}^{d}$
Tom 103 / 2005
Colloquium Mathematicum 103 (2005), 231-239
MSC: Primary 28A80.
DOI: 10.4064/cm103-2-8
Streszczenie
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.
