New York Journal of Mathematics
Volume 20 (2014) 1253-1268

  

Arthur Baragar

Lattice points on hyperboloids of one sheet

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Published: December 12, 2014
Keywords: Gauss' circle problem, lattice points, orbits, Hausdorff dimension, ample cone
Subject: 11D45, 11P21, 20H10, 22E40, 11N45, 14J28, 11G50, 11H06

Abstract
The problem of counting lattice points on a hyperboloid of two sheets is Gauss' circle problem in hyperbolic geometry. The problem of counting lattice points on a hyperboloid of one sheet does not have the same geometric interpretation, and in general, the solution(s) to Gauss' circle problem gives a lower bound, but not an upper bound. In this paper, we describe an exception. Given an ample height, and a lattice on a hyperboloid of one sheet generated by a point in the interior of the effective cone, the problem can be reduced to Gauss' circle problem.

Acknowledgements

This work is based upon research supported by the National Security Agency under grant H98230-08-1-0022.


Author information

Department of Mathematical Sciences, University of Nevada, Las Vegas, NV 89154-4020
baragar@unlv.nevada.edu