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The multifractal spectrum of discrete measures. (English) Zbl 0789.28007
Let \(\mu\) be any measure of the unit interval such that each subinterval has positive measure. The lower local dimension of \(\mu\) is defined as \[ d_ \mu(x)=\liminf_{\varepsilon\to 0} {\log \mu(U_ \varepsilon(x))\over \log\varepsilon}, \] where \(U_ \varepsilon(x)\) is the open \(\varepsilon\)-neighbourhood of \(x\). Inspired by an example due to Kahane and Katznelson the authors suggest to replace the usual definition of the \(f(\alpha)\)-spectrum by \[ f(\alpha)=\lim_{\varepsilon\to 0}\dim\{x\mid \alpha-\varepsilon\leq d_ \mu(x)\leq \alpha+\varepsilon\},\quad 0\leq \alpha\leq\infty, \] where dim denotes Hausdorff dimension. They construct then a discrete measure \(\mu\) such that \(f(\alpha)\) becomes a linear function.

MSC:
28A80 Fractals
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