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Examples on harmonic measure and normal numbers


Author: Jang-Mei Wu
Journal: Proc. Amer. Math. Soc. 95 (1985), 211-216
MSC: Primary 30C85; Secondary 31A15
DOI: https://doi.org/10.1090/S0002-9939-1985-0801325-X
MathSciNet review: 801325
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Abstract: Suppose that $ F$ is a bounded set in $ {{\mathbf{R}}^m}$, $ m \geqslant 2$, with positive capacity. Add to $ F$ a disjoint set $ E$ so that $ E \cup F$ is closed, and let $ D = {{\mathbf{R}}^m}\backslash (E \cup F)$. Under what conditions on the added set $ E$ do we have harmonic measure $ \omega (F,D) = 0$? It turns out that besides the size of $ E$ near $ F$, the location of $ E$ relative to $ F$ also plays an important role. Our example, based on normal numbers, stresses this fact.


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DOI: https://doi.org/10.1090/S0002-9939-1985-0801325-X
Article copyright: © Copyright 1985 American Mathematical Society

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