Hausdorff dimension of $\tau$-approximable points on self-similar sets in ${\mathbb{R}}^d$

Abstract

Let $d\ge 1$. Let $K\subset {\mathbb{R}}^d$ be a non-singleton self-similar set generated by a finite strongly irreducible iterated function system satisfying the open set condition, and let $\delta ={\dim }_{\mathrm{H}}K$. For $\tau >1/d$, set \[ W_d(τ) = \left\{ \mathbf{x}\in\mathbb{R}^d: |q\mathbf{x}-\mathbf{p}|<q^{-τ} \text{ for infinitely many }(\mathbf{p},q)\in\mathbb{Z}^d\times\mathbb{N} \right\}. \] We prove that there exists ${\epsilon }_K>0$ such that, for every $1/d<\tau <1/d+{\epsilon }_K$, \[ \mathcal{H}^{s(τ)}(K\cap W_d(τ))=\infty, \qquad\text{with } s(τ):=δ+\frac{d+1}{1+τ}-d, \] and consequently \[ \dim_{\mathrm H}(K\cap W_d(τ)) = δ+\frac{d+1}{1+τ}-d. \] In dimension one, specializing to the middle-third Cantor set, this establishes the Bugeaud--Durand conjectural formula for $\tau >1$ sufficiently close to $1$.