A note on the distance set problem in the plane
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Abstract:
We use a simple geometric-combinatorial argument to establish a quantitative relation between the generalized Hausdorff measure of a set and its distance set, extending a result originally due to Falconer.References
- Lennart Carleson, Selected problems on exceptional sets, Van Nostrand Mathematical Studies, No. 13, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto, Ont.-London, 1967. MR 0225986
- K. J. Falconer, On the Hausdorff dimensions of distance sets, Mathematika 32 (1985), no. 2, 206–212 (1986). MR 834490, DOI 10.1112/S0025579300010998
- 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, DOI 10.1112/S0025579300013462
- Thomas Wolff, Decay of circular means of Fourier transforms of measures, Internat. Math. Res. Notices 10 (1999), 547–567. MR 1692851, DOI 10.1155/S1073792899000288
Additional Information
- Themis Mitsis
- Affiliation: Department of Mathematics and Statistics, University of Jyväskylä, P.O. Box 35, FIN-40351 Jyväskylä, Finland
- Address at time of publication: Nestou 6, Athens 14342, Greece
- Email: mitsis@math.jyu.fi, tmitsis@yahoo.com
- Received by editor(s): November 21, 2000
- Published electronically: October 12, 2001
- Additional Notes: This research has been supported by a Marie Curie Fellowship of the European Community programme “Improving human potential and the socio-economic knowledge base" under contract number HPMFCT-2000-00442.
- Communicated by: David Preiss
- © Copyright 2001 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 130 (2002), 1669-1672
- MSC (2000): Primary 28A12, 28A78
- DOI: https://doi.org/10.1090/S0002-9939-01-06375-4
- MathSciNet review: 1887013