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

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 $s\in (0,1\mathrm{/}2)$, there exists an $s$-Ahlfors regular compact set $K\subset \mathbb{R}\subset \mathbb{C}$ such that $\mathrm{d}\mathrm{i}\mathrm{m}{\omega }^{\mathrm{\infty }}=s$, where ${\omega }^{\mathrm{\infty }}$ 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 $1\mathrm{/}2$ is sharp.