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 | References | Similar Articles | Additional Information

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:
https://doi.org/10.1090/S0002-9939-04-07603-8

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