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 $0<\tau \le 1$, where $W_2(\tau )=\{x\in [0,1]:\mathrm{\parallel }{2}^{n}x\mathrm{\parallel }<n^{-\tau }\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{i}\mathrm{n}\mathrm{f}\mathrm{i}\mathrm{n}\mathrm{i}\mathrm{t}\mathrm{e}\mathrm{l}\mathrm{y}\mathrm{m}\mathrm{a}\mathrm{n}\mathrm{y}n\in \mathbb{N}\}$. We prove that the conjecture holds for $\tau >\frac{1}{\gamma }-\frac{1-\gamma }{3-\gamma }(\approx 1.429)$ and $0<\tau <\frac{\gamma }{12}(\approx 0.052)$, where $\gamma =\frac{\log 2}{\log 3}$ 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 ($0<\tau \le 0.01$, due to Baker (2025)). Our key innovation is to establish the estimate $$\sum _{n=1}^N\mathrm{\mid }\widehat{\mu }(h{2}^n){\mathrm{\mid }}^2\ll N^{1-\gamma }$$ and its consequences: $$\sum _{n=1}^N\mathrm{\mid }\widehat{\mu }(h{2}^n)\mathrm{\mid }\ll N^{1-\frac{\gamma }{2}},\sum _{n=1}^N{n}^{-\sigma }\mathrm{\mid }\widehat{\mu }(h{2}^n)\mathrm{\mid }{\ll }_{\sigma }{N}^{1-\frac{\gamma }{2}-\sigma },$$ 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.