One-Ring Bourgain Alternative And A 5-Decimal Bound On The Dimension Of Harmonic Measure

Abstract

Improving theorems of Bourgain and Badger--Genschaw, we prove that the Hausdorff dimension of harmonic measure on any domain in $\mathbb{R}^3$, $\mathbb{R}^4$, and $\mathbb{R}^n$ for $n\geq3$ is at most $3-10^{-5}$, $4-10^{-7}$, and $n-(n-1)^{-(2n+6.12\sqrt{n})}$, respectively. The previous bounds in dimensions $3$ and $4$ were $3-10^{-15}$ and $4-10^{-26}$, while Bishop conjectures that the sharp bound is $n-(n-1)^{-1}$. The main difficulty in optimizing Bourgain's method is estimating non-additive $s$-dimensional Hausdorff-type contents when $s<n$ is close to $n$. Previous arguments used net contents associated to a fixed $m$-adic lattice, with $m$ large depending on $n$. We redevelop the dimension theory using pairwise disjoint cubes in general position and obtain a new estimate showing that one thin ring without large holes surrounding a cube makes it unlikely for Brownian motion to reach the cube before exiting the domain. A key feature of the construction is that the holes are round rather than rectangular. We also propose ten challenges on the dimension of harmonic measure that may provide accessible targets en route to Bishop's conjecture.