Besicovitch subsets of self-similar sets
\textit{M. Morán} and \textit{J.-M. Rey} [Trans. Am. Math. Soc. 350, 2297-2310 (1998; Zbl 0899.28002)] defined self-similar Besicovitch sets -- after \textit{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 [\textit{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 [\textit{M. Morán} and \textit{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 [\textit{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 \textit{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.
scientific article; zbMATH DE number 1794824