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Large sets of zero analytic capacity. (English) Zbl 1062.30025
The authors consider a Cantor type set in the unit square and prove that it has zero analytic capacity but it does not have \(\sigma\)-finite one-dimensional Hausdorff measure. The first example of such a Cantor set was given by L. D. Ivanov [On sets of analytic capacity zero, in: Linear and Complex Analysis Problem Book 3, Part II (V.P. Khavin, S.V. Kruschev, N.K. Nikolskii, ed.), Lecture Notes Math. 1043, 498–501 (1984; Zbl 0545.30038)]. The proof in the paper under review is based on some estimates for harmonic measure and Green function due to P. W. Jones [Lecture Notes Math. 1384, 24–68 (1989; Zbl 0675.30029)] and a stopping-time argument. The result is related to a conjecture of P. Mattila [Publ. Math., Barc. 40, No. 1, 195–204 (1996; Zbl 0888.30026)] on the characterization of Cantor sets with zero analytic capacity.

30C85 Capacity and harmonic measure in the complex plane
28A75 Length, area, volume, other geometric measure theory
28A78 Hausdorff and packing measures
31A15 Potentials and capacity, harmonic measure, extremal length and related notions in two dimensions
Full Text: DOI
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