Abstract.
We show that for any \(\varepsilon > 0\) and any region G whose outer boundary equals \(\{z: |z| = 1\}\), there is a sequence \(\{\Delta_n\}_{n=1}^{\infty}\) of pairwise disjoint closed disks in G such that \(\{z: |z| = 1\}\) is the set of accumulation points of \(\{\Delta_n\}_{n=1}^{\infty}, \sum_{n=1}^{\infty}radius(\Delta_n) < \varepsilon\) and \(\omega_{\Omega}\) (harmonic measure on the boundary of \(\Omega := \{z:|z| < 1\}\setminus (\cup_{n=1}^{\infty}\Delta_n)\) for evaluation at some \(z_o\) in \(\Omega\)) is supported on \(\cup_{n=1}^{\infty}(\partial\Delta_n)\).
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Received: 10 March 2000 / Revised version: 6 November 2001/Published online: 28 February 2002
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Akeroyd, J. Champagne subregions of the disk whose bubbles carry harmonic measure. Math Ann 323, 267–279 (2002). https://doi.org/10.1007/s002080100303
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DOI: https://doi.org/10.1007/s002080100303