Modified Shrinking Target Problem for Matrix Transformations of Tori

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.