Jarník-type theorem for self-similar sets

Let $K\subset {\mathbb{R}}^d$ be a compact subset equipped with a $\delta$-Ahlfors regular measure $\mu$. For any $\tau >1\mathrm{/}d$ and any inhomogeneous'' vector $\boldsymbolθ\in\mathbb R^d$, let $W_d(ψ_τ,\boldsymbolθ)$ denote the set of $(ψ_τ,\boldsymbolθ)$-well approximable numbers, where $ψ_τ(q)=q^{-τ}$. Assuming a local estimate for the $μ$-measure of the intersections of $K$ with the neighborhoods of rational’’ vectors $(\mathbf{p}+\mathit{\theta })\mathrm{/}q$, we establish a sharp upper bound for the Hausdorff dimension of $K\cap W_d({\psi }_{\tau },\mathit{\theta })$, 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 Bénard, He and Zhang [arXiv:2508.09076]. We also provide some non-trivial missing digits set $K\subset [0,1{]}^d$ whose intersection with $W_d({\psi }_{\tau },0)$ has full $(\delta +\frac{1+d}{1+\tau }-d)$-Hausdorff measure.