Examples on harmonic measure and normal numbers
HTML articles powered by AMS MathViewer
- by Jang-Mei Wu PDF
- Proc. Amer. Math. Soc. 95 (1985), 211-216 Request permission
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.References
- A. S. Besicovitch, On the sum of digits of real numbers represented in the dyadic system, Math. Ann. 110 (1935), no. 1, 321–330. MR 1512941, DOI 10.1007/BF01448030
- J. L. Doob, Classical potential theory and its probabilistic counterpart, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 262, Springer-Verlag, New York, 1984. MR 731258, DOI 10.1007/978-1-4612-5208-5
- G. H. Hardy and E. M. Wright, An introduction to the theory of numbers, Oxford, at the Clarendon Press, 1954. 3rd ed. MR 0067125
- Peter W. Jones, A geometric localization theorem, Adv. in Math. 46 (1982), no. 1, 71–79. MR 676987, DOI 10.1016/0001-8708(82)90054-8 S. Kaczmarz and H. Steinhaus, Theorie der Orthogonalreihen, PWN, Warsaw, 1935.
- Robert Kaufman, A further example on scales of Hausdorff functions, J. London Math. Soc. (2) 8 (1974), 585–586. MR 357721, DOI 10.1112/jlms/s2-8.4.585
- Robert Kaufman and Jang Mei Wu, Distortion of the boundary under conformal mapping, Michigan Math. J. 29 (1982), no. 3, 267–280. MR 674280
- Bernt Øksendal, Brownian motion and sets of harmonic measure zero, Pacific J. Math. 95 (1981), no. 1, 179–192. MR 631668
- Jang-Mei Wu, On singularity of harmonic measure in space, Pacific J. Math. 121 (1986), no. 2, 485–496. MR 819202
Additional Information
- © Copyright 1985 American Mathematical Society
- 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