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