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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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