Diophantine approximations and directional discrepancy of rotated lattices

In this paper we study the following question related to Diophantine approximations and geometric measure theory: for a given set Ω find α such that α − θ has bad Diophantine properties simultaneously for all θ ∈ Ω. How do the arising Diophantine inequalities depend on the geometry of the set Ω? We provide several methods which yield different answers in terms of the metric entropy of Ω and consider various examples. Furthermore, we apply these results to explore the asymptotic behavior of the directional discrepancy, i.e., the discrepancy with respect to rectangles rotated in certain sets of directions. It is well known that the extremal cases of this problem (fixed direction vs. all possible rotations) yield completely different bounds. We use rotated lattices to obtain directional discrepancy estimates for general rotation sets and investigate the sharpness of these methods.