Examples on harmonic measure and normal numbers

Wu, Jang-Mei

doi:10.1090/S0002-9939-1985-0801325-X

Examples on harmonic measure and normal numbers
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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

Theorie der Orthogonalreihen

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