Lower bounds on Bourgain's constant for harmonic measure

TL;DR

This paper establishes new lower bounds on Bourgain's constant for harmonic measure in higher dimensions, showing it decreases at a specific rate as dimension increases, and provides explicit estimates for dimensions 3 and 4.

Contribution

It improves the known lower bounds on Bourgain's constant for all dimensions greater than or equal to 3, refining previous results and providing explicit numerical estimates.

Findings

Proved that b_n ≥ c n^{-2n(n-1)}/ln(n) for all n ≥ 3.

Estimated b_3 ≥ 10^{-15} and b_4 ≥ 2×10^{-26}.

Showed Bourgain's constant decreases at a specific rate with increasing dimension.

Abstract

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 $b_4\geq 2\times 10^{-26}$.