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

Abstract

Let $C$ be the middle-third Cantor set and $\hat{I}¼$ be the Cantor-Lebesgue measure on $C$. A conjecture of Velani states that $\hat{I}¼(W_2(\ddot{I}„))=0$ if $\ddot{I}„>1$ and $\hat{I}¼(W_2(\ddot{I}„))=1$ if $0<\ddot{I}„\le 1$, where $W_2(\ddot{I}„)=\left\{x\in [0,1]:\Vert 2^{n}x\Vert <n^{-\ddot{I}„}\mathrm{for}\mathrm{infinitely}\mathrm{many}n\in \mathbb{N}\right\}$. We prove that the conjecture holds for $\ddot{I}„>{\frac{1}{\hat{I}}}^3-\frac{1-{\hat{I}}^3}{3-{\hat{I}}^3}(\approx 1.429)$ and $0<\ddot{I}„<\frac{\hat{I}}{³}12(\approx 0.052)$, where ${\hat{I}}^3=\frac{\log 2}{\log 3}$ is the Hausdorff dimension of $C$. This improves the known results on both the null part ($\ddot{I}„>{\frac{1}{\hat{I}}}^3-\frac{0.078(1-{\hat{I}}^3)}{{\hat{I}}^3(2-{\hat{I}}^3)}\approx 1.552$, due to Allen, Baker, Chow, and Yu (2023)) and the full measure part ($0<\ddot{I}„\le 0.01$, 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 $0<\ddot{I}ƒ<1-\frac{\hat{I}}{³}2$, 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.

31 pages