Jarn\'ik-type theorem for self-similar sets

Abstract

Let $K\subset\mathbb R^d$ be a compact subset equipped with a $\delta$-Ahlfors regular measure $\mu$. For any $\tau>1/d$ and any ``inhomogeneous'' vector $\boldsymbol{\theta}\in\mathbb R^d$, let $W_d(\psi_\tau,\boldsymbol{\theta})$ denote the set of $(\psi_\tau,\boldsymbol{\theta})$-well approximable numbers, where $\psi_\tau(q)=q^{-\tau}$. Assuming a local estimate for the $\mu$-measure of the intersections of $K$ with the neighborhoods of ``rational'' vectors $(\mathbf p+\boldsymbol{\theta})/q$, we establish a sharp upper bound for the Hausdorff dimension of $K\cap W_d(\psi_\tau,\boldsymbol{\theta})$, together with some nontrivial lower bounds when $\tau$ is below a certain threshold. One of the lower bounds becomes sharp in the one-dimensional homogeneous case ($d=1$, $\theta=0$) for a class of sufficiently thick self-similar sets $K$, and moreover $K\cap W_1(\psi_\tau,0)$ has full $(\delta+\frac{2}{1+\tau}-1)$-Hausdorff measure. These results have several applications: (1) the set of homogeneous very well approximable numbers has full Hausdorff dimension within strongly irreducible self-similar sets in $\mathbb R^d$, extending a recent result of Chen [arXiv:2510.17096]; (2) the set of inhomogeneous very well approximable numbers has full Hausdorff dimension within sufficiently thick missing digits sets in $\mathbb R$, affirmatively answering a question posed by Yu [arXiv:2101.05910]. Our applications build on the seminal works of Yu [arXiv:2101.05910] and B\'enard, He and Zhang [arXiv:2508.09076]. We also provide some non-trivial missing digits set $K\subset[0,1]^d$ whose intersection with $W_d(\psi_\tau,0)$ has full $(\delta+\frac{1+d}{1+\tau}-d)$-Hausdorff measure.