Jarník-type theorem for self-similar sets

Abstract

Let $K\subset {\mathbb{R}}^d$ be a compact subset equipped with a $\hat{I}´$-Ahlfors regular measure $\hat{I}¼$. For any $\ddot{I}„>1/d$ and any ``inhomogeneous'' vector $\hat{\mathbf{I}}¸\in {\mathbb{R}}^d$, let $W_d(\ddot{I}{ˆ}_{\ddot{I}}„,\hat{\mathbf{I}}¸)$ denote the set of $(\ddot{I}{ˆ}_{\ddot{I}}„,\hat{\mathbf{I}}¸)$-well approximable numbers, where $\ddot{I}{ˆ}_{\ddot{I}}„(q)=q^{-\ddot{I}„}$. Assuming a local estimate for the $\hat{I}¼$-measure of the intersections of $K$ with the neighborhoods of ``rational'' vectors $(\mathbf{p}+\hat{\mathbf{I}}¸)/q$, we establish a sharp upper bound for the Hausdorff dimension of $K\cap W_d(\ddot{I}{ˆ}_{\ddot{I}}„,\hat{\mathbf{I}}¸)$, together with some nontrivial lower bounds when $\ddot{I}„$ is below a certain threshold. One of the lower bounds becomes sharp in the one-dimensional homogeneous case ($d=1$, $\hat{I}¸=0$) for a class of sufficiently thick self-similar sets $K$, and moreover $K\cap W_1(\ddot{I}{ˆ}_{\ddot{I}}„,0)$ has full $(\hat{I}´+\frac{2}{1+\ddot{I}„}-1)$-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 Bénard, He and Zhang [arXiv:2508.09076]. We also provide some non-trivial missing digits set $K\subset [0,1{]}^d$ whose intersection with $W_d(\ddot{I}{ˆ}_{\ddot{I}}„,0)$ has full $(\hat{I}´+\frac{1+d}{1+\ddot{I}„}-d)$-Hausdorff measure.

Include several non-trivial self-similar sets whose intersections with the $\ddot{I}„$-approximable set have the desired Hausdorff dimension