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Diophantine approximations and directional discrepancy of rotated lattices


Authors: Dmitriy Bilyk, Xiaomin Ma, Jill Pipher and Craig Spencer
Journal: Trans. Amer. Math. Soc. 368 (2016), 3871-3897
MSC (2010): Primary 11K38, 11K60, 28A78, 52C05
DOI: https://doi.org/10.1090/tran/6492
Published electronically: September 9, 2015
MathSciNet review: 3453360
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Abstract: 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 \in \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.


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Additional Information

Dmitriy Bilyk
Affiliation: School of Mathematics, University of Minnesota, Minneapolis, Minnesota 55455
Email: dbilyk@math.umn.edu

Xiaomin Ma
Affiliation: Department of Mathematics, Brown University, Providence, Rhode Island 02912
Email: xiaomin@math.brown.edu

Jill Pipher
Affiliation: Department of Mathematics, Brown University, Providence, Rhode Island 02912
Email: jpipher@math.brown.edu

Craig Spencer
Affiliation: Department of Mathematics, Kansas State University, Manhattan, Kansas 66506
Email: cvs@math.ksu.edu

DOI: https://doi.org/10.1090/tran/6492
Received by editor(s): February 27, 2013
Received by editor(s) in revised form: March 24, 2014
Published electronically: September 9, 2015
Article copyright: © Copyright 2015 American Mathematical Society