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

Abstract

Let $d\geq 1$. 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 $δ=\dim_{\mathrm H} K$. For $τ>1/d$, set \[ W_d(τ) = \left\{ \mathbf{x}\in\mathbb{R}^d: |q\mathbf{x}-\mathbf{p}| 0$ such that, for every $1/d<τ<1/d+\varepsilon_K$, \[ \mathcal{H}^{s(τ)}(K\cap W_d(τ))=\infty, \qquad\text{with } s(τ):=δ+\frac{d+1}{1+τ}-d, \] 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 $τ>1$ sufficiently close to $1$.