Metric results for dyadic approximation on the middle-third Cantor set

Abstract

Let $C$ be the middle-third Cantor set and $μ$ be the Cantor-Lebesgue measure on $C$. A conjecture of Velani states that $μ(W_2(τ))=0$ if $τ>1$ and $μ(W_2(τ))=1$ if $0<τ\leq 1$, where $W_2(τ)=\left\{x\in[0,1]: \|2^nx\|<n^{-τ}\ {\rm for\ infinitely\ many }\ n\in\mathbb{N} \right\}$. We prove that the conjecture holds for $τ>\frac{1}γ-\frac{1-γ}{3-γ}\,(\approx 1.429)$ and $0<τ<\fracγ{12}\,(\approx 0.052)$, where $γ=\frac{\log2}{\log3}$ is the Hausdorff dimension of $C$. This improves the known results on both the null part ($τ>\frac{1}γ-\frac{0.078(1-γ)}{γ(2-γ)}\approx 1.552$, due to Allen, Baker, Chow, and Yu (2023)) and the full measure part ($0<τ\leq 0.01$, due to Baker (2025)). Our key innovation is to establish the estimate \[\sum_{n=1}^{N}|\widehatμ(h2^n)|^2\ll N^{1-γ}\] and its consequences: \[ \sum_{n=1}^{N}|\widehatμ(h2^n)|\ll N^{1-\fracγ{2}},\quad \sum_{n=1}^{N}n^{- σ}|\widehatμ(h2^n)|\ll_σ N^{1-\fracγ{2}-σ},\] where $0<σ<1-\fracγ{2}$, 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.