Let E ⊂ ℝn+1, n ≥ 2, be a uniformly rectifiable set of dimension n. Then bounded harmonic functions in Ω := ℝn+1 \ E satisfy Carleson measure estimates and are ε-approximable. Our results may be viewed as generalized versions of the classical F. and M. Riesz theorem, since the estimates that we prove are equivalent, in more topologically friendly settings, to quantitative mutual absolute continuity of harmonic measure and surface measure.
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Acknowledgments. Hofmann's work was partially supported by National Science Foundation (NSF) grant DMS-1361701. Martell's work was supported in part by Ministerio de Econom?a y Competitividad grant MTM2010-16518, Instituto de Ciencias Matem?ticas Severo Ochoa project SEV-2011-0087, and the European Research Council (ERC) under the European Union's Seventh Framework Programme (FP7/2007-2013)/ERC agreement no. 615112 HAPDEGMT. Mayboroda's work was partially supported by an Alfred P. Sloan Fellowship, the NSF CAREER award DMS-1056004, the NSF INSPIRE award DMS-1344235, and the NSF Materials Research Science and Engineering Center seed grant DMR-0212302.