On shrinking targets for linear expanding and hyperbolic toral endomorphisms

Let A be an invertible d× d matrix with integer elements. Then A determines a self-map T of the d-dimensional torus Tᵈ=Rᵈ/Zᵈ. Given a real number τ>0, and a sequence ₙ\ of points in Tᵈ, let W_τ be the set of points xᵈ such that Tⁿ(x)∈ B(zₙ,e-nτ) for infinitely many n. The Hausdorff dimension of W_τ has previously been studied by Hill--Velani and Li--Liao--Velani--Zorin. We provide complete results on the Hausdorff dimension of W_τ for any expanding matrix. For hyperbolic matrices, we compute the dimension of W_τ only when A is a 2 × 2 matrix. We give counterexamples to a natural candidate for a dimension formula for general dimension d.