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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Finding integral diagonal pairs in a two dimensional $ \mathcal{N}$-set


Authors: Lev A. Borisov and Renling Jin
Journal: Proc. Amer. Math. Soc. 139 (2011), 2431-2434
MSC (2010): Primary 11B75, 11H06, 11P21
Published electronically: December 20, 2010
MathSciNet review: 2784808
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Abstract | References | Similar Articles | Additional Information

Abstract: According to Nathanson, an $ n$-dimensional $ \mathcal{N}$-set is a compact subset $ A$ of $ \mathbb{R}^n$ such that for every $ x\in\mathbb{R}^n$ there is $ y\in A$ with $ y-x\in\mathbb{Z}^n$. We prove that every two dimensional $ \mathcal{N}$-set $ A$ must contain distinct points $ x,y$ such that $ x-y$ is in $ \mathbb{Z}^2$ and $ x-y$ is neither horizontal nor vertical. This answers a question of P. Hegarty and M. Nathanson.


References [Enhancements On Off] (What's this?)

  • 1. M.B. Nathanson, An inverse problem in number theory and geometric group theory, in ``Additive Number Theory'', ed. D. Chudnovsky and G. Chudnovsky, Springer, New York, 2010, pp. 249-258.
  • 2. Z. Ljujic, C. Sanabria, A note on the inverse problem for the lattice points, arXiv:1006.5740v1 [math.NT], 29 June 2010.

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Additional Information

Lev A. Borisov
Affiliation: Department of Mathematics, Rutgers University, Piscataway, New Jersey 08854

Renling Jin
Affiliation: Department of Mathematics, College of Charleston, Charleston, South Carolina 29424

DOI: http://dx.doi.org/10.1090/S0002-9939-2010-10688-3
PII: S 0002-9939(2010)10688-3
Received by editor(s): July 7, 2010
Published electronically: December 20, 2010
Additional Notes: The work of the first author was partially supported by NSF Grant 1003445
The work of the second author was partially supported by NSF Grant RUI 0500671.
Communicated by: Ken Ono
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.