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

L. Olsen

Characterization of local dimension functions of subsets of

ℝ^{d}

L. Olsen

Colloquium Mathematicae, Tome 103 (2005), p. 231-239 / Harvested from The Polish Digital Mathematics Library

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Résumé

For a subset $E \subseteq ℝ^{d}$ and $x \in ℝ^{d}$ , the local Hausdorff dimension function of E at x is defined by $d i m_{H, l o c} (x, E) = l i m_{r ↘ 0} d i m_{H} (E \cap B (x, r))$ where $d i m_{H}$ 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.

Publié le : 2005-01-01

Zbl 1105.28007

EUDML-ID : urn:eudml:doc:284206

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L. Olsen. Characterization of local dimension functions of subsets of $ℝ^{d}$
            . Colloquium Mathematicae, Tome 103 (2005) pp. 231-239. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm103-2-8/