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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 >0, and a sequence \zₙ\ of points in Tᵈ, let W_ be the set of points xᵈ such that Tⁿ (x) B (zₙ, e^-n) for infinitely many n. The Hausdorff dimension of W_ has previously been studied by Hill--Velani and Li--Liao--Velani--Zorin. We provide complete results on the Hausdorff dimension of W_ for any expanding matrix. For hyperbolic matrices, we compute the dimension of W_ only when A is a 2 2 matrix. We give counterexamples to a natural candidate for a dimension formula for general dimension d.
Hu et al. (Sat,) studied this question.