Year: 2026
Type: preprint
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. $\sum_{k} |\hatμ(2^k)|^σ< \infty$ for any $σ> 0$, where $μ$ is the Cantor measure.
Source: arXiv (Cornell University)
Authors: Chow, Sam, Yu, Han
Institutions: University of Warwick, Chongqing University
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7223206276