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

Hofmann, S.; Iosevich, A.

doi:10.1090/S0002-9939-04-07603-8

			Journals Home Search Author Info Subscribe Tech Support My Subscriptions Browser Compatibility Info Help
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
Posted: July 22, 2004
MathSciNet review: 2085162
Full-text PDF Free Access

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.

[Arutyunyants04] G. Arutyunyants, Geometric methods in harmonic analysis, Doctoral Dissertation (in preparation) (2004).
[Bourgain94] Jean Bourgain, Hausdorff dimension and distance sets, Israel J. Math. 87 (1994), no. 1-3, 193–201. MR 1286826 (95h:28008), http://dx.doi.org/10.1007/BF02772994
[Falconer86] K. J. Falconer, On the Hausdorff dimensions of distance sets, Mathematika 32 (1985), no. 2, 206–212 (1986). MR 834490 (87j:28008), http://dx.doi.org/10.1112/S0025579300010998
[Herz62] C. S. Herz, Fourier transforms related to convex sets, Ann. of Math. (2) 75 (1962), 81–92. MR 0142978 (26 #545)
[IoLa2003] A. Iosevich and I. $\L$ aba, -distance sets and Falconer conjecture, (in preparation) (2002).
[Kahane68] Jean-Pierre 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] Pertti Mattila, Spherical averages of Fourier transforms of measures with finite energy; dimension of intersections and distance sets, Mathematika 34 (1987), no. 2, 207–228. MR 933500 (90a:42009), http://dx.doi.org/10.1112/S0025579300013462
[PaAg95] János Pach and Pankaj K. Agarwal, Combinatorial geometry, Wiley-Interscience Series in Discrete Mathematics and Optimization, John Wiley & Sons Inc., New York, 1995. A Wiley-Interscience Publication. MR 1354145 (96j:52001)
[PeresSchlag00] Yuval Peres and Wilhelm Schlag, Smoothness of projections, Bernoulli convolutions, and the dimension of exceptions, Duke Math. J. 102 (2000), no. 2, 193–251. MR 1749437 (2001d:42013), http://dx.doi.org/10.1215/S0012-7094-00-10222-0
[Sogge93] Christopher D. Sogge, Fourier integrals in classical analysis, Cambridge Tracts in Mathematics, vol. 105, Cambridge University Press, Cambridge, 1993. MR 1205579 (94c:35178)
[Solomyak98] Boris Solomyak, Measure and dimension for some fractal families, Math. Proc. Cambridge Philos. Soc. 124 (1998), no. 3, 531–546. MR 1636589 (99e:28016), http://dx.doi.org/10.1017/S0305004198002680
[Sjolin93] Per Sjölin, Estimates of spherical averages of Fourier transforms and dimensions of sets, Mathematika 40 (1993), no. 2, 322–330. MR 1260895 (95f:28007), http://dx.doi.org/10.1112/S0025579300007087
[Wolff99] Thomas Wolff, Decay of circular means of Fourier transforms of measures, Internat. Math. Res. Notices 10 (1999), 547–567. MR 1692851 (2000k:42016), http://dx.doi.org/10.1155/S1073792899000288
[Wolff02] Thomas H. Wolff, Lectures on harmonic analysis, University Lecture Series, vol. 29, American Mathematical Society, Providence, RI, 2003. With a foreword by Charles Fefferman and preface by Izabella Łaba; Edited by Łaba and Carol Shubin. MR 2003254 (2004e:42002)

	Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
\|