Lower bounds on Bourgain's constant for harmonic measure

For every $n\ge 2$, 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 $n\ge 3$, Bourgain (1987) proved that $b_n>0$ and Wolff (1995) produced examples showing $b_n<1$. Refining Bourgain’s original outline, we prove that $$b_n\ge c{n}^{-2n(n-1)}\mathrm{/}\ln (n)$$ for all $n\ge 3$, where $c>0$ is a constant that is independent of $n$. We further estimate $b_3\ge 1\times {10}^{-15}$ and $b_4\ge 2\times {10}^{-26}$.