Metric results for dyadic approximation on the middle-third Cantor set
Let $C$ be the middle-third Cantor set and $\mu$ be the Cantor-Lebesgue measure on $C$. A conjecture of Velani states that $\mu(W_2(\tau))=0$ if $\tau>1$ and $\mu(W_2(\tau))=1$ if $0<\tau\leq 1$, where $W_2(\tau)=\left\{x\in[0,1]: \|2^nx\| \frac{1}{\gamma}-\frac{1-\gamma}{3-\gamma}\,(\approx 1.429)$ and $0<\tau<\frac{\gamma}{12}\,(\approx 0.052)$, where $\gamma=\frac{\log2}{\log3}$ is the Hausdorff dimension of $C$. This improves the known results on both the null part ($\tau>\frac{1}{\gamma}-\frac{0.078(1-\gamma)}{\gamma(2-\gamma)}\approx 1.552$, due to Allen, Baker, Chow, and Yu (2023)) and the full measure part ($0<\tau\leq 0.01$, due to Baker (2025)). Our key innovation is to establish the estimate \[\sum_{n=1}^{N}|\widehat{\mu}(h2^n)|^2\ll N^{1-\gamma}\] and its consequences: \[ \sum_{n=1}^{N}|\widehat{\mu}(h2^n)|\ll N^{1-\frac{\gamma}{2}},\quad \sum_{n=1}^{N}n^{- \sigma}|\widehat{\mu}(h2^n)|\ll_{\sigma} N^{1-\frac{\gamma}{2}-\sigma},\] where $0<\sigma<1-\frac{\gamma}{2}$, and all estimates are uniform in $h\in\mathbb{Z}\setminus\{0\}$. For the full measure part, our approach also generalizes to self-similar measures on a class of missing-digit sets.