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Besicovitch subsets of self-similar sets. (English) Zbl 1024.28005
M. Morán and J.-M. Rey [Trans. Am. Math. Soc. 350, 2297-2310 (1998; Zbl 0899.28002)] defined self-similar Besicovitch sets – after A. S. Besicovitch [Math. Ann. 110, 321-330 (1894; Zbl 0009.39503)] – as the sets of points of a self-similar set with prescribed frequencies \(p_i\) in their generating similarities \(\varphi_i\) (\(i=1,2,\ldots,m)\). In the case that the “open set” separation condition holds, they showed that the Hausdorff and packing dimensions of Besicovitch sets are given by \(\alpha= {\sum_{i=1}^{m}p_i\log p_i \over \sum_{i=1}^{m}p_i\log r_i}\), where \(r_i\) is the contraction ratio of \(\varphi_i\) – so generalizing the Besicovitch-Eggleston formula [H. G. Eggleston, Quart. J. Math., Oxford Ser. 20, 31-36 (1949; Zbl 0031.20801)]; disregard the only exceptional case that \(\alpha\) coincides with the Hausdorff dimension of the self-similar set. They also showed that the \(\alpha\)-dimensional Hausdorff measure of Besicovitch sets should be either zero or infinity, suggested infinity as the correct value [op.cit.], and proved elsewhere that their \(\alpha\)-dimensional packing measure is infinity [M. Morán and J.-M. Rey, Ann. Acad. Sci. Fenn., Math. 22, 365-386 (1997; Zbl 0890.28005)].
In the paper under review, the authors prove that the dichotomy zero – infinity in fact occurs for any gauge function \(g\) used to compute any Hausdorff or packing measure of Besicovitch sets. The key fact that Hausdorff measure is \(+\infty\) for any function gauging dimension \(\alpha\) is proved – following an idea used by Peres to show that self-affine carpets have infinite Hausdorff measure [Y. Peres, Math. Proc. Cambr. Philos. Soc. 116, 513-526 (1994; Zbl 0811.28005)] – by deforming the natural Bernoulli measure associated with the probability vector \((p_i)\). Their main theorem extends the pioneering work by R. Kaufman [J. Lond. Math. Soc., II. Ser. 8, 585-586 (1974; Zbl 0302.28015)]. In particular, their result confirms the conjecture of Morán and Rey.

MSC:
28A80 Fractals
28A78 Hausdorff and packing measures
11K55 Metric theory of other algorithms and expansions; measure and Hausdorff dimension
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References:
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