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

TL;DR

This paper establishes sharp bounds on the Hausdorff dimension of intersections between self-similar sets and sets of well-approximable points, extending classical Diophantine approximation results to fractal sets.

Contribution

It provides new upper and lower bounds for Hausdorff dimensions of Diophantine approximation sets within self-similar fractals, including sharp results in the one-dimensional case.

Findings

Full Hausdorff dimension of homogeneous very well approximable numbers in certain self-similar sets.

Full Hausdorff dimension of inhomogeneous very well approximable numbers in thick missing digits sets.

Construction of missing digits sets with full Hausdorff measure in the approximation set.

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.