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Falconer estimate for circular and cinematic Furstenberg problem
Abstract
We prove the Falconer-type bound dimHF≥ u+1 for planar circular (u,v)-Furstenberg sets whenever 0<u≤1, 0<v≤3, and 2u+v≥3. The same bound holds for smooth cinematic curve families. This resolves the circular Furstenberg conjecture of Green, Harris, Ou, Ren, and Tammen in its full Falconer range 2u+v≥3.
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BibTeX
@misc{xie2026falconer,
title = {{Falconer estimate for circular and cinematic Furstenberg problem}},
author = {Xiang Xie},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.35213},
eprint = {2610.35213},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.35213},
}
RIS
TY - UNPB
TI - Falconer estimate for circular and cinematic Furstenberg problem
AU - Xiang Xie
PY - 2026
DA - 2026/10/26
PB - arXiv
DO - 10.48550/arxiv.2610.35213
UR - https://arxiv.org/abs/2610.35213
AB - We prove the Falconer-type bound dim_HF≥ u+1 for planar circular (u,v)-Furstenberg sets whenever 0<u≤1, 0<v≤3, and 2u+v≥3. The same bound holds for smooth cinematic curve families. This resolves the circular Furstenberg conjecture of Green, Harris, Ou, Ren, and Tammen in its full Falconer range 2u+v≥3.
ER -