For s-Ahlfors regular sets with β2 non-flatness at least δ0, harmonic measure dimension drops below s when s exceeds n minus c δ0 squared.
Volberg,On the dimension of harmonic measure of Cantor repellers, Michigan Math
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Dimension Drop for Harmonic Measure on Ahlfors Regular Boundaries
For s-Ahlfors regular sets with β2 non-flatness at least δ0, harmonic measure dimension drops below s when s exceeds n minus c δ0 squared.