The Besicovitch sets have infinite Hausdorff measures (Q2719682)
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scientific article; zbMATH DE number 1609866
| Language | Label | Description | Also known as |
|---|---|---|---|
| English | The Besicovitch sets have infinite Hausdorff measures |
scientific article; zbMATH DE number 1609866 |
Statements
25 June 2001
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Besicovitch set
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Hausdorff measure
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Hausdorff dimension
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0.8788724
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0.87077487
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0.87002736
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0.86811256
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0.86774015
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The Besicovitch sets have infinite Hausdorff measures (English)
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In this paper, the author proves that the Besicovitch set has infinite Hausdorff measure.NEWLINENEWLINENEWLINEFor a fixed probability vector \(p= \{p_1,p_2,\dots, p_{m-1}\}\), the Besicovitch set is defined as follows NEWLINE\[NEWLINEB= \Biggl\{x\in [0,1]: \lim_{n\to \infty} {1\over n} \sum^n_{k=1} \tau_j(x_k)= p_j,\;0\leq j\leq m-1\Biggr\},NEWLINE\]NEWLINE where \(x\) is in \(m\)-adic expansion. As we know, the Hausdorff dimension of \(B\) is NEWLINE\[NEWLINE\dim(B)= {-\sum p_j\log p_j\over\log m},NEWLINE\]NEWLINE and the author proves that the Besicovitch set has infinite Hausdorff measure in the above dimension \(\dim(B)\).
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