HAUSDORFF DIMENSION OF THE CARTESIAN PRODUCT OF LIMSUP SETS IN DIOPHANTINE APPROXIMATION

The metric theory of limsup sets is the main topic in metric Diophantine approximation. A very simple observation by Erdos shows the dimension of the Cartesian product of two sets of Liouville numbers is 1. To disclose the mystery hidden there, we consider and present a general principle for the Hausdorff dimension of the Cartesian product of limsup sets. As an application of our general principle, it is found that dim(H )W(psi) x<middle dot><middle dot><middle dot> x W (psi) = d - 1 + dim(H )W(psi) where W (psi) is the set of psi-well approximable points in R and psi : N -> R+ is a positive non-increasing function. Even this concrete case was never observed before. Our result can also be compared with Marstrand's famous inequality on the dimension of the Cartesian product of general sets.