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 ℝ3, ℝ4, and ℝn for n≥3 is at most 3−10−5, 4−10−7, and n−(n−1)−(2n+6.12√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.
Related
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- Failure of dimension drop for harmonic measure on Ahlfors regular flat boundaries
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BibTeX
@misc{albert2026onering,
title = {{One-Ring Bourgain Alternative And A 5-Decimal Bound On The Dimension Of Harmonic Measure}},
author = {Michael Albert and Matthew Badger and Alyssa Genschaw},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.28037},
eprint = {2610.28037},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.28037},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - One-Ring Bourgain Alternative And A 5-Decimal Bound On The Dimension Of Harmonic Measure
AU - Michael Albert
AU - Matthew Badger
AU - Alyssa Genschaw
PY - 2026
DA - 2026/10/20
PB - arXiv
DO - 10.48550/arxiv.2610.28037
UR - https://arxiv.org/abs/2610.28037
AB - Improving theorems of Bourgain and Badger--Genschaw, we prove that the Hausdorff dimension of harmonic measure on any domain in ℝ^3, ℝ^4, and ℝ^n for n≥3 is at most 3−10^−5, 4−10^−7, and n−(n−1)^−(2n+6.12√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.
ER -