Distance measures for well-distributed sets

A Iosevich, M Rudnev

Research output: Contribution to journalArticle (Academic Journal)peer-review

9 Citations (Scopus)


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.
Translated title of the contributionDistance measures for well-distributed sets
Original languageEnglish
Pages (from-to)61 - 80
Number of pages20
JournalDiscrete and Computational Geometry
Volume38 (1)
Publication statusPublished - Jul 2007

Bibliographical note

Publisher: Springer

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