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.
Bibliographical noteFunding Information:
The authors are deeply grateful to the American Institute of Mathematics for their hospitality - most of this research has been carried out during their participation in the SQuaREs program. The authors would also like to thank the National Science Foundation for support: grants DMS 0801036, 1101519 (first author), DMS 0901139 (third and second authors). The fourth author was also supported by NSA Young Investigator Grants #H98230-10-1-0155 and #H98230-12-1-0220. In addition, the authors are indebted to William Chen, Giancarlo Travaglini, and Nikolay Moshchevitin for numerous interesting and fruitful discussions.