Discrete & Computational Geometry

, Volume 38, Issue 1, pp 61–80

Distance Measures for Well-Distributed Sets

Article

DOI: 10.1007/s00454-007-1316-9

Cite this article as:
Iosevich, A. & Rudnev, M. Discrete Comput Geom (2007) 38: 61. doi:10.1007/s00454-007-1316-9

Abstract

In this paper we investigate the Erdos/Falconer distance conjecture for a natural class of sets statistically, though not necessarily arithmetically, similar to a lattice. We prove a good upper bound for spherical means that have been classically used to study this problem. We conjecture that a majorant for the spherical means suffices to prove the distance conjecture(s) in this setting. For a class of non-Euclidean distances, we show that this generally cannot be achieved, at least in dimension two, by considering integer point distributions on convex curves and surfaces. In higher dimensions, we link this problem to the question about the existence of smooth well-curved hypersurfaces that support many integer points.

Copyright information

© Springer 2007

Authors and Affiliations

  1. 1.Department of Mathematics, University of MissouriColumbia, MO 65211USA
  2. 2.Department of Mathematics, University of BristolBristol BS8 1TWEngland