科研之友
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摘要
In this paper we study the following question related to Diophantine approximations and geometric measure theory: for a given set Omega find alpha such that alpha - theta has bad Diophantine properties simultaneously for all theta is an element of Omega. How do the arising Diophantine inequalities depend on the geometry of the set Omega? We provide several methods which yield different answers in terms of the metric entropy of Omega 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.
- 出版日期2016-6
