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PreprintCC BY 4.0arXiv

Failure of dimension drop for harmonic measure on Ahlfors regular flat boundaries

  • Yingying Cai
  • Cole Jeznach
  • Xavier Tolsa
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Abstract

We develop an N-adic modification of the construction of David-Jeznach-Julia for compact subsets of the real line ℝ⊂ℂ for which harmonic measure for the complementary domains has no dimension drop. In particular, we show that for every s∈(0,1/2), there exists an s-Ahlfors regular compact set K⊂ℝ⊂ℂ such that dimω∞=s, where ω∞ denotes harmonic measure for ℂ∖ K with pole at infinity. Combined with a result of the third author, this shows that the threshold 1/2 is sharp.

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Cite

BibTeXRIS
BibTeX
@misc{cai2026failure,
  title = {{Failure of dimension drop for harmonic measure on Ahlfors regular flat boundaries}},
  author = {Yingying Cai and Cole Jeznach and Xavier Tolsa},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.15805},
  eprint = {2610.15805},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.15805},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Failure of dimension drop for harmonic measure on Ahlfors regular flat boundaries
AU  - Yingying Cai
AU  - Cole Jeznach
AU  - Xavier Tolsa
PY  - 2026
DA  - 2026/10/11
PB  - arXiv
DO  - 10.48550/arxiv.2610.15805
UR  - https://arxiv.org/abs/2610.15805
AB  - We develop an N-adic modification of the construction of David-Jeznach-Julia for compact subsets of the real line ℝ⊂ℂ for which harmonic measure for the complementary domains has no dimension drop. In particular, we show that for every s∈(0,1/2), there exists an s-Ahlfors regular compact set K⊂ℝ⊂ℂ such that dimω^∞=s, where ω^∞ denotes harmonic measure for ℂ∖ K with pole at infinity. Combined with a result of the third author, this shows that the threshold 1/2 is sharp.
ER  -