LARGE INTERSECTION PROPERTIES FOR LIMSUP SETS GENERATED BY RECTANGLES IN COMPACT METRIC SPACES

Abstract

The large intersection classes in real Euclidean spaces have been introduced by Falconer in 1985. First, we generalize Falconer’s large intersection classes to compact metric spaces and show that these classes are closed under countable intersections. Next, we prove that the limsup sets generated by rectangles in a compact metric space enjoy this intersection property under certain full Hausdorff measure assumptions. We also apply our results to high-dimensional Diophantine approximation on fractals.