On shrinking targets for linear expanding and hyperbolic toral endomorphisms

Abstract Let be an invertible matrix with integer elements. Then determines a self‐map of the ‐dimensional torus . Given a real number , and a sequence of points in , let be the set of points such that for infinitely many . The Hausdorff dimension of has previously been studied by Hill–Velani and Li–Liao–Velani–Zorin. We provide a lower bound on the Hausdorff dimension of for any expanding matrix. For hyperbolic matrices, we compute the dimension of only when is a matrix. We give counterexamples to a natural candidate for a dimension formula for general dimension .