PreprintCC BY 4.0arXiv
On a conjecture of Velani
Abstract
We investigate the approximation of elements of the middle-third Cantor set by dyadic rationals, and natural extensions. We prove a conjecture of Velani, completing the metric theory for the Cantor measure. Our key method enables a quantitative study of the incommensurability of binary and ternary expansions of integers, initiated by Furstenberg. The method has other applications, e.g. ∑k |μ̂(2k)|σ< ∞ for any σ> 0, where μ is the Cantor measure.
Related
BibTeX
@misc{chow2026conjecture,
title = {{On a conjecture of Velani}},
author = {Sam Chow and Han Yu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.28839},
eprint = {2610.28839},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.28839},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - On a conjecture of Velani
AU - Sam Chow
AU - Han Yu
PY - 2026
DA - 2026/10/20
PB - arXiv
DO - 10.48550/arxiv.2610.28839
UR - https://arxiv.org/abs/2610.28839
AB - We investigate the approximation of elements of the middle-third Cantor set by dyadic rationals, and natural extensions. We prove a conjecture of Velani, completing the metric theory for the Cantor measure. Our key method enables a quantitative study of the incommensurability of binary and ternary expansions of integers, initiated by Furstenberg. The method has other applications, e.g. ∑_k |μ̂(2^k)|^σ< ∞ for any σ> 0, where μ is the Cantor measure.
ER -