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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Diophantine approximations and directional discrepancy of rotated lattices
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by Dmitriy Bilyk, Xiaomin Ma, Jill Pipher and Craig Spencer PDF
Trans. Amer. Math. Soc. 368 (2016), 3871-3897 Request permission

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
  • 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
  • © Copyright 2015 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 368 (2016), 3871-3897
  • MSC (2010): Primary 11K38, 11K60, 28A78, 52C05
  • DOI: https://doi.org/10.1090/tran/6492
  • MathSciNet review: 3453360