Lower bounds on Bourgain's constant for harmonic measure

For every $n\geq 2$, Bourgain's constant $b_n$ is the largest number such that the (upper) Hausdorff dimension of harmonic measure is at most $n-b_n$ for every domain in $\mathbb{R}^n$ on which harmonic measure is defined. Jones and Wolff (1988) proved that $b_2=1$. When $n\geq 3$, Bourgain (1987) proved that $b_n>0$ and Wolff (1995) produced examples showing $b_n<1$. Refining Bourgain's original outline, we prove that $$b_n\geq c\,n^{-2n(n-1)}/\ln(n)$$ for all $n\geq 3$, where $c>0$ is a constant that is independent of $n$. We further estimate $b_3\geq 1\times 10^{-15}$ and $b_4\geq 2\times 10^{-26}$.