Abstract
In this paper, we calculate the Hausdorff dimension of the fractal set [Formula: see text] where [Formula: see text] is the standard [Formula: see text]-transformation with [Formula: see text], [Formula: see text] is a positive function on [Formula: see text] and [Formula: see text] is the usual metric on the torus [Formula: see text]. Moreover, we investigate a modified version of the shrinking target problem, which unifies the shrinking target problems and quantitative recurrence properties for matrix transformations of tori. Let [Formula: see text] be a [Formula: see text] non-singular matrix with real coefficients. Then, [Formula: see text] determines a self-map of the [Formula: see text]-dimensional torus [Formula: see text]. For any [Formula: see text], let [Formula: see text] be a positive function on [Formula: see text] and [Formula: see text] with [Formula: see text]. We obtain the Hausdorff dimension of the fractal set [Formula: see text] where [Formula: see text] is a hyperrectangle and [Formula: see text] is a sequence of Lipschitz vector-valued functions on [Formula: see text] with a uniform Lipschitz constant.
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