Failure of dimension drop for harmonic measure on Ahlfors regular flat boundaries

Abstract

We develop an $N$-adic modification of the construction of David-Jeznach-Julia for compact subsets of the real line $\mathbb{R}\subset\mathbb{C}$ for which harmonic measure for the complementary domains has no dimension drop. In particular, we show that for every $s\in(0,1/2)$, there exists an $s$-Ahlfors regular compact set $K\subset\mathbb{R}\subset\mathbb{C}$ such that $\mathrm{dim}ω^\infty=s$, where $ω^\infty$ denotes harmonic measure for $\mathbb{C}\setminus K$ with pole at infinity. Combined with a result of the third author, this shows that the threshold $1/2$ is sharp.