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

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 $\tau >1\mathrm{/}d$, set $$W_d(\tau )=\{\mathbf{x}\in {\mathbb{R}}^d:\mathrm{\mid }q\mathbf{x}-\mathbf{p}\mathrm{\mid }<q^{-\tau }\text{ for infinitely many }(\mathbf{p},q)\in {\mathbb{Z}}^d\times \mathbb{N}\}\mathrm{.}$$ We prove that there exists ${\epsilon }_K>0$ such that, for every $1\mathrm{/}d<\tau <1\mathrm{/}d+{\epsilon }_K$, $${\mathcal{H}}^{s(\tau )}(K\cap W_d(\tau ))=\mathrm{\infty },\text{with }s(\tau ):=\delta +\frac{d+1}{1+\tau }-d,$$ and consequently $${\dim }_{\mathrm{H}}(K\cap W_d(\tau ))=\delta +\frac{d+1}{1+\tau }-d\mathrm{.}$$ In dimension one, specializing to the middle-third Cantor set, this establishes the Bugeaud–Durand conjectural formula for $\tau >1$ sufficiently close to $1$.