Proceedings of the Edinburgh Mathematical Society, 2005
For a subset $E\subseteq\mathbb{R}^{d}$ and $x\in\mathbb{R}^{d}$, the local Hausdorff dimension f... more 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$$ \mathrm{dim}_{\mathrm{loc}}(x,E)=\lim_{r\searrow0}\mathrm{dim}(E\cap B(x,r)), $$where ‘dim’ denotes the Hausdorff dimension. Using some of our earlier results on so-called multifractal divergence points we give a short proof of the following result: any continuous function $f:\mathbb{R}^{d}\to[0,d]$ is the local dimension function of some set $E\subseteq\mathbb{R}^{d}$. In fact, our result also provides information about the rate at which the dimension $\mathrm{dim}(E\cap B(x,r))$ converges to $f(x)$ as $r\searrow0$.AMS 2000 Mathematics subject classification: Primary 28A80
Proceedings of the Edinburgh Mathematical Society, 2005
For a subset $E\subseteq\mathbb{R}^{d}$ and $x\in\mathbb{R}^{d}$, the local Hausdorff dimension f... more 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$$ \mathrm{dim}_{\mathrm{loc}}(x,E)=\lim_{r\searrow0}\mathrm{dim}(E\cap B(x,r)), $$where ‘dim’ denotes the Hausdorff dimension. Using some of our earlier results on so-called multifractal divergence points we give a short proof of the following result: any continuous function $f:\mathbb{R}^{d}\to[0,d]$ is the local dimension function of some set $E\subseteq\mathbb{R}^{d}$. In fact, our result also provides information about the rate at which the dimension $\mathrm{dim}(E\cap B(x,r))$ converges to $f(x)$ as $r\searrow0$.AMS 2000 Mathematics subject classification: Primary 28A80
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