Champagne subregions of the unit disc (Q2846752)

Let \(B\) denote the open unit disc and \(E\) be the union of a sequence of pairwise disjoint closed discs \(\overline{B}(x_{k},r_{k})\) such that \(\left| x_{k}\right| \rightarrow 1\) and \(\sup_{k}r_{k}/(1-\left| x_{k}\right| )<1\). The set \(E\) is called unavoidable (for Brownian motion) if it carries full harmonic measure with respect to the domain \(\Omega =B\backslash E\). The author obtains quite precise results about which configurations of discs are unavoidable. For example, Theorem 1.1 contains the following two statements. (a) If \(E\) is unavoidable, then NEWLINE\[NEWLINE\sum_{k}\frac{ (1-\left| x_{k}\right| )^{2}}{\left| y-x_{k}\right| ^{2}} \left\{ \log \frac{1-\left| x_{k}\right| }{r_{k}}\right\} ^{-1}=\infty NEWLINE\]NEWLINE for almost every \(y\in \partial B\). (b) Conversely, if the above condition holds and, further, NEWLINE\[NEWLINE\inf_{j\neq k}\frac{\left| x_{k}-x_{j}\right| }{1-\left| x_{k}\right| }\left\{ \log \frac{ 1-\left| x_{k}\right| }{r_{k}}\right\} ^{1/2}>0,NEWLINE\]NEWLINE then \(E\) is unavoidable. Corresponding results in higher dimensions had previously been obtained by the reviewer and \textit{M. Ghergu} [Ann. Acad. Sci. Fenn., Math. 35, No. 1, 321--329 (2010; Zbl 1203.31007)], but the two-dimensional situation required separate investigation. Theorem 1.1 leads to refinements of results of \textit{J. R. Akeroyd} [Math. Ann. 323, No.2, 267--279 (2002; Zbl 1006.30019)] and of \textit{J. Ortega-Cerdà} and \textit{K. Seip} [Indiana Univ. Math. J. 53, No. 3, 905--923 (2004; Zbl 1057.30022)].