On shrinking targets for linear expanding and hyperbolic toral endomorphisms

Let $A$ be an invertible $d\times d$ matrix with integer elements. Then $A$ determines a self-map $T$ of the $d$-dimensional torus ${\mathbb{T}}^d={\mathbb{R}}^d\mathrm{/}{\mathbb{Z}}^d$. Given a real number $\tau >0$, and a sequence $\{z_n\}$ of points in ${\mathbb{T}}^d$, let $W_{\tau }$ be the set of points $x\in {\mathbb{T}}^d$ such that $T^n(x)\in B(z_n,e^{-n\tau })$ for infinitely many $n\in \mathbb{N}$. The Hausdorff dimension of $W_{\tau }$ has previously been studied by Hill–Velani and Li–Liao–Velani–Zorin. We provide complete results on the Hausdorff dimension of $W_{\tau }$ for any expanding matrix. For hyperbolic matrices, we compute the dimension of $W_{\tau }$ only when $A$ is a $2\times 2$ matrix. We give counterexamples to a natural candidate for a dimension formula for general dimension $d$.