- Current Issue - Ahead of Print Articles - All Issues - Search - Open Access - Information for Authors - Downloads - Guideline - Regulations ㆍPaper Submission ㆍPaper Reviewing ㆍPublication and Distribution - Code of Ethics - For Authors ㆍOnlilne Submission ㆍMy Manuscript - For Reviewers - For Editors
 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.2015.52.5.1489Printed 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 Downloads: Full-text PDF