Besicovitch subsets of self-similar sets.

*(English)*Zbl 1024.28005M. 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.

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.

Reviewer: José-Manuel Rey (Madrid)

##### MSC:

28A80 | Fractals |

28A78 | Hausdorff and packing measures |

11K55 | Metric theory of other algorithms and expansions; measure and Hausdorff dimension |

##### References:

[1] | On the sum of digits of real numbers represented in the dyadic system, Math. Ann, 110, 321-330, (1934) · JFM 60.0949.01 |

[2] | The fractional dimension of a set defined by decimal properties, Quart. J. Math. Oxford Ser, 20, 31-36, (1949) · Zbl 0031.20801 |

[3] | Techniques in Fractal Geometry, (1997), John Wiley and sons inc. · Zbl 0869.28003 |

[4] | A further example on scales of Hausdorff functions, J. London Math. Soc, 8, 2, 585-586, (1974) · Zbl 0302.28015 |

[5] | Singularity of self-similar measures with respect to Hausdorff measures, Trans. of Amer. Math. Soc., 350, 6, 2297-2310, (1998) · Zbl 0899.28002 |

[6] | The self-affine carpets of McMullen and Bedford have infinite Hausdorff measure, Math. Proc. Camb. Phil. Soc, 116, 513-526, (1994) · Zbl 0811.28005 |

[7] | Probability, (1984), Springer-Verlag, New York · Zbl 0536.60001 |

[8] | The measure theory of random fractals, Math. Proc. Cambridge Philo. Soc, 100, 383-408, (1986) · Zbl 0622.60021 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.