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