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
Full-text PDF Free Access
Abstract | References | Similar Articles | Additional Information
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
MR Author ID:
757936
Email:
dbilyk@math.umn.edu
Xiaomin Ma
Affiliation:
Department of Mathematics, Brown University, Providence, Rhode Island 02912
MR Author ID:
960706
Email:
xiaomin@math.brown.edu
Jill Pipher
Affiliation:
Department of Mathematics, Brown University, Providence, Rhode Island 02912
MR Author ID:
237541
Email:
jpipher@math.brown.edu
Craig Spencer
Affiliation:
Department of Mathematics, Kansas State University, Manhattan, Kansas 66506
MR Author ID:
867353
Email:
cvs@math.ksu.edu
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