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 $T^d=R^d/Z^d$. Given a real number $\tau >0$, and a sequence $\left\{z_n\right\}$ of points in $T^d$, let $W_{\tau }$ be the set of points $x\in T^d$ such that $T^n\left(x\right)\in B(z_n,e^{-n\tau })$ for infinitely many $n\in N$. The Hausdorff dimension of $W_{\tau }$ has previously been studied by Hill–Velani and Li–Liao–Velani–Zorin. We provide a lower bound 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 $d$.