On simultaneous local dimension functions of subsets of $\mathbb R^{d}$
Bull. Korean Math. Soc. 2015 Vol. 52, No. 5, 1489-1493
https://doi.org/10.4134/BKMS.2015.52.5.1489
Printed September 30, 2015
Lars Olsen
University of St. Andrews
Abstract : For a subset $E\subseteq\mathbb R^{d}$ and $x\in\mathbb R^{d}$, the local Hausdorff dimension function of $E$ at $x$ and the local packing dimension function of $E$ at $x$ are defined by \[\begin{aligned} \dim_{\Haus,\loc}(x,E) &= \lim_{r\searrow0}\dim_{\Haus}(E\cap B(x,r))\,,\\ \dim_{\Pack,\loc}(x,E) &= \lim_{r\searrow0}\dim_{\Pack}(E\cap B(x,r))\,, \end{aligned} \] where $\dim_{\Haus}$ and $\dim_{\Pack}$ denote the Hausdorff dimension and the packing dimension, respectively. In this note we give a short and simple proof showing that for any pair of continuous functions $f,g:\mathbb R^{d}\to[0,d]$ with $f\le g$, it is possible to choose a set $E$ that simultaneously has $f$ as its local Hausdorff dimension function and $g$ as its local packing dimension function.
Keywords : Hausdorff dimension, packing dimension, local Hausdorff dimension, local packing dimension
MSC numbers : 28A80
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