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arXiv · 2610.28839

On a conjecture of Velani

Sam Chow·Han Yu
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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.

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Sam Chow, Han Yu. 2026-10-20. On a conjecture of Velani. https://arxiv.org/abs/2610.28839

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