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

 

Circular averages and Falconer/Erdös distance conjecture in the plane for random metrics


Authors: S. Hofmann and A. Iosevich
Journal: Proc. Amer. Math. Soc. 133 (2005), 133-143
MSC (2000): Primary 42B10, 52C10
Published electronically: July 22, 2004
MathSciNet review: 2085162
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Abstract: We study a variant of the Falconer distance problem for perturbations of the Euclidean and related metrics. We prove that Mattila's criterion, expressed in terms of circular averages, which would imply the Falconer conjecture, holds on average. We also use a technique involving diophantine approximation to prove that the well-distributed case of the Erdös Distance Conjecture holds for almost every appropriate perturbation of the Euclidean metric.


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Additional Information

S. Hofmann
Affiliation: Department of Mathematics, University of Missouri, Columbia, Missouri 65211
Email: hofmann@math.missouri.edu

A. Iosevich
Affiliation: Department of Mathematics, University of Missouri, Columbia, Missouri 65211
Email: iosevich@math.missouri.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-04-07603-8
PII: S 0002-9939(04)07603-8
Received by editor(s): July 19, 2003
Received by editor(s) in revised form: September 8, 2003
Published electronically: July 22, 2004
Additional Notes: The research of the authors was partially supported by NSF grants
Communicated by: Andreas Seeger
Article copyright: © Copyright 2004 American Mathematical Society