Finding integral diagonal pairs in a two dimensional $\mathcal {N}$–set
HTML articles powered by AMS MathViewer
- by Lev A. Borisov and Renling Jin PDF
- Proc. Amer. Math. Soc. 139 (2011), 2431-2434 Request permission
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
- 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.
- Z. Ljujic, C. Sanabria, A note on the inverse problem for the lattice points, arXiv:1006.5740v1 [math.NT], 29 June 2010.
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
- 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
- © Copyright 2010
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 139 (2011), 2431-2434
- MSC (2010): Primary 11B75, 11H06, 11P21
- DOI: https://doi.org/10.1090/S0002-9939-2010-10688-3
- MathSciNet review: 2784808