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), 133143
MSC (2000):
Primary 42B10, 52C10
Published electronically:
July 22, 2004
MathSciNet review:
2085162
Fulltext PDF Free Access
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References 
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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 welldistributed case of the Erdös Distance Conjecture holds for almost every appropriate perturbation of the Euclidean metric.
 [Arutyunyants04]
G. Arutyunyants, Geometric methods in harmonic analysis, Doctoral Dissertation (in preparation) (2004).
 [Bourgain94]
J. Bourgain, Hausdorff dimension and distance sets, Israel J. Math. 87 (1994), 193201. MR 1286826 (95h:28008)
 [Falconer86]
K. J. Falconer, On the Hausdorff dimensions of distance sets, Mathematika 32 (1986), 206212. MR 0834490 (87j:28008)
 [Herz62]
C. Herz, Fourier transforms related to convex sets, Ann. of Math. 75 (1962), 8192. MR 0142978 (26:545)
 [IoLa2003]
A. Iosevich and I. aba, distance sets and Falconer conjecture, (in preparation) (2002).
 [Kahane68]
J. P. Kahane, Some random series of functions, D. C. Heath and Co. Raytheon Education Co., Lexington, Mass. (1968). MR 0254888 (40:8095)
 [KatzTardos03]
N. Katz and G. Tardos, On the Erdös Distance Problem, (preprint) (2003).
 [Mattila87]
P. Mattila, Spherical averages of Fourier transforms of measures with finite energy: dimensions of intersections and distance sets, Mathematika 34 (1987), 207228. MR 0933500 (90a:42009)
 [PaAg95]
J. Pach and and P. Agarwal, Combinatorial Geometry, WileyInterscience Series (1995). MR 1354145 (96j:52001)
 [PeresSchlag00]
Y. Peres and W. Schlag, Smoothness of projections, Bernoulli convolutions, and the dimension of exceptions, Duke Math. J. 102 (2000), 193251. MR 1749437 (2001d:42013)
 [Sogge93]
C. D. Sogge, Fourier integrals in classical analysis, Cambridge University Press (1993). MR 1205579 (94c:35178)
 [Solomyak98]
B. Solomyak, Measure and dimension for some fractal families, Math. Proc. Cambridge Philos. Soc. 124 (1998), 531546. MR 1636589 (99e:28016)
 [Sjolin93]
P. Sjölin, Estimates of spherical averages of Fourier transforms and dimensions of sets, Mathematika 40 (1993), 322330. MR 1260895 (95f:28007)
 [Wolff99]
T. Wolff, Decay of circular means of Fourier transforms of measures, Internat. Math. Res. Notices 10 (1999), 547567. MR 1692851 (2000k:42016)
 [Wolff02]
T. Wolff, Lectures on Harmonic Analysis, California Institute of Technology Class Lectures Notes (revised by I. aba) http://www.math.ubc.ca (2002). University Lecture Series, no. 29, Amer. Math. Soc. (2003). MR 2003254 (2004e:42002)
 [Arutyunyants04]
 G. Arutyunyants, Geometric methods in harmonic analysis, Doctoral Dissertation (in preparation) (2004).
 [Bourgain94]
 J. Bourgain, Hausdorff dimension and distance sets, Israel J. Math. 87 (1994), 193201. MR 1286826 (95h:28008)
 [Falconer86]
 K. J. Falconer, On the Hausdorff dimensions of distance sets, Mathematika 32 (1986), 206212. MR 0834490 (87j:28008)
 [Herz62]
 C. Herz, Fourier transforms related to convex sets, Ann. of Math. 75 (1962), 8192. MR 0142978 (26:545)
 [IoLa2003]
 A. Iosevich and I. aba, distance sets and Falconer conjecture, (in preparation) (2002).
 [Kahane68]
 J. P. Kahane, Some random series of functions, D. C. Heath and Co. Raytheon Education Co., Lexington, Mass. (1968). MR 0254888 (40:8095)
 [KatzTardos03]
 N. Katz and G. Tardos, On the Erdös Distance Problem, (preprint) (2003).
 [Mattila87]
 P. Mattila, Spherical averages of Fourier transforms of measures with finite energy: dimensions of intersections and distance sets, Mathematika 34 (1987), 207228. MR 0933500 (90a:42009)
 [PaAg95]
 J. Pach and and P. Agarwal, Combinatorial Geometry, WileyInterscience Series (1995). MR 1354145 (96j:52001)
 [PeresSchlag00]
 Y. Peres and W. Schlag, Smoothness of projections, Bernoulli convolutions, and the dimension of exceptions, Duke Math. J. 102 (2000), 193251. MR 1749437 (2001d:42013)
 [Sogge93]
 C. D. Sogge, Fourier integrals in classical analysis, Cambridge University Press (1993). MR 1205579 (94c:35178)
 [Solomyak98]
 B. Solomyak, Measure and dimension for some fractal families, Math. Proc. Cambridge Philos. Soc. 124 (1998), 531546. MR 1636589 (99e:28016)
 [Sjolin93]
 P. Sjölin, Estimates of spherical averages of Fourier transforms and dimensions of sets, Mathematika 40 (1993), 322330. MR 1260895 (95f:28007)
 [Wolff99]
 T. Wolff, Decay of circular means of Fourier transforms of measures, Internat. Math. Res. Notices 10 (1999), 547567. MR 1692851 (2000k:42016)
 [Wolff02]
 T. Wolff, Lectures on Harmonic Analysis, California Institute of Technology Class Lectures Notes (revised by I. aba) http://www.math.ubc.ca (2002). University Lecture Series, no. 29, Amer. Math. Soc. (2003). MR 2003254 (2004e:42002)
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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/S0002993904076038
PII:
S 00029939(04)076038
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
