A class of sets with positive length and zero analytic capacity. (English) Zbl 0592.30028

Mattila, Pertti

Ann. Acad. Sci. Fenn., Ser. A I, Math. 10, 387-395 (1985).

Examples of compact plane sets with positive length and zero analytic capacity were first given by Vitushkin (1959) and Garnett (1970). In this article it is shown that a compact plane set E has zero analytic capacity provided that: (i) E has finite 1-dimensional Hausdorff measure (i.e. \(H^ 1(E)<\infty)\). (ii) E is irregular (in the sense of Besicovitch). (iii) The lower Besicovitch density D(E,x) is positive, for \(H^ 1\) a.a. \(x\in E\). (iv) For \(H^ 1\) a.a. \(x\in E\) there exists a direction which is not a weak tangent direction for E at x. The examples of Vitushkin and Garnett satisfy these conditions, in addition to a large class of self- similar sets.

Reviewer: B.Øksendal

MSC:

30C85	Capacity and harmonic measure in the complex plane
28A12	Contents, measures, outer measures, capacities
28A75	Length, area, volume, other geometric measure theory

Keywords:

zero analytic capacity; 1-dimensional Hausdorff measure; weak tangent direction