The Hausdorf And Packing Dimensions Of Some Sets Related To High-dimensional Sierpinski Carpets

The Sierpinski carpets first considered by C.McMullen and later studied by Y.peres are modified by insisting that the allowed digits in the expansions occur with prescribed frequencies. A.Nielsen calculates the Hausdorff, parking, and box dimensions of the modified Sierpinski carpets. Based on this basis, we consider High-dimensional Sierpinski carpets and this paper (l)calculates the Hausdorff, parking, and box dimensions of its frequency set; (2)calculates the Hausdorff dimensions of its projection set. But this is not simple expansions, which needs a new way to construct the product measure of closed rectangles. And the estimation of inequalities is also quite complex. So we do not give necessary and sufficient conditions for these measures to be infinite.