Failure of dimension drop for harmonic measure on Ahlfors regular boundaries of dimension close to $1$

Abstract

For every $0<s<1$ we construct a compact $s$-Ahlfors regular set $K\subset \mathbb{C}$ for which harmonic measure of $\mathbb{C}\setminus K$ with pole at infinity is comparable to ${\mathcal{H}}^s{|}_K$. In particular, there is no dimension drop for any dimension $s<1$. The construction is $N$-adic and based on configurations close to the $N$-th roots of unity. We adjust the positions of the children by small polynomial deformations so that the logarithmic potential of the natural Bernoulli measure becomes constant in the limit. A first-mode correction gives the range $s<2/3$, while corrections of arbitrary finite order yield the full range $0<s<1$.