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Failure of dimension drop for harmonic measure on Ahlfors regular boundaries of dimension close to $1$

  • Yingying Cai
  • Cole Jeznach
  • Xavier Tolsa
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Abstract

For every 0<s<1 we construct a compact s-Ahlfors regular set K⊂ℂ for which harmonic measure of ℂ∖ K with pole at infinity is comparable to ℋs|K. In particular, there is no dimension drop for any dimension s<1. The construction is N-adic and based on configurations close to the N-th roots of unity. We adjust the positions of the children by small polynomial deformations so that the logarithmic potential of the natural Bernoulli measure becomes constant in the limit. A first-mode correction gives the range s<2/3, while corrections of arbitrary finite order yield the full range 0<s<1.

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Cite

BibTeXRIS
BibTeX
@misc{cai2026failure,
  title = {{Failure of dimension drop for harmonic measure on Ahlfors regular boundaries of dimension close to $1$}},
  author = {Yingying Cai and Cole Jeznach and Xavier Tolsa},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.15759},
  eprint = {2610.15759},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.15759},
}
RIS
TY  - UNPB
TI  - Failure of dimension drop for harmonic measure on Ahlfors regular boundaries of dimension close to $1$
AU  - Yingying Cai
AU  - Cole Jeznach
AU  - Xavier Tolsa
PY  - 2026
DA  - 2026/10/11
PB  - arXiv
DO  - 10.48550/arxiv.2610.15759
UR  - https://arxiv.org/abs/2610.15759
AB  - For every 0<s<1 we construct a compact s-Ahlfors regular set K⊂ℂ for which harmonic measure of ℂ∖ K with pole at infinity is comparable to ℋ^s|_K. In particular, there is no dimension drop for any dimension s<1. The construction is N-adic and based on configurations close to the N-th roots of unity. We adjust the positions of the children by small polynomial deformations so that the logarithmic potential of the natural Bernoulli measure becomes constant in the limit. A first-mode correction gives the range s<2/3, while corrections of arbitrary finite order yield the full range 0<s<1.
ER  -