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On the Hausdorff dimensions of distance sets

Part of: Classical measure theory

Published online by Cambridge University Press:  26 February 2010

K. J. Falconer
K. J. Falconer
Affiliation:
School of Mathematics, University Walk, Bristol, BS8 1TW
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Extract

If E is a subset of ℝn (n ≥ 1) we define the distance set of E as

The best known result on distance sets is due to Steinhaus [11], namely, that, if E ⊂ ℝn is measurable with positive n-dimensional Lebesgue measure, then D(E) contains an interval [0, ε) for some ε > 0. A number of variations of this have been examined, see Falconer [6, p. 108] and the references cited therein.

MSC classification

Secondary: 28A75: Length, area, volume, other geometric measure theory
Type
Research Article
Information
Mathematika , Volume 32 , Issue 2 , December 1985 , pp. 206 - 212
Copyright
Copyright © University College London 1985

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References

1.Carleson, L.. Selected Problems on Exceptional Sets (Van Nostrand, Princeton, 1967).Google Scholar
2.Chung, F. R. K.. The number of different distances determined by n points in the plane J. Combinatorial Theory A, 26 (1984), 342–354.CrossRefGoogle Scholar
3.Davies, Roy O.. Subsets of finite measure in analytic sets. Indag. Math., 14 (1952), 488–489.Google Scholar
4.Davies, Roy O.. Two counter-examples concerning Hausdorff dimensions of projections. Colloq. Math., 42 (1979), 53–58.Google Scholar
5.Davies, Roy O.. Rings of dimension d. To appear.Google Scholar
6.Falconer, K. J.. The Geometry of Fractal Sets (Cambridge University Press, 1984).Google Scholar
7.Falconer, K. J.. Rings of fractional dimension. Mathematika, 31 (1984), 25–27.CrossRefGoogle Scholar
8.Falconer, K. J.. Classes of sets with large intersection. Mathematika, 32 (1985), 191–205.Google Scholar
9.Marstrand, J. M.. The dimension of cartesian product sets. Proc. Cambridge Phil. Soc., 50 (1954), 198–202.CrossRefGoogle Scholar
10.Rogers, C. A.. Hausdorff Measures (Cambridge University Press, 1970).Google Scholar
11.Steinhaus, H.. Sur les distances des points des ensembles de mesure positive. Fund. Math., 1 (1920), 93–104.CrossRefGoogle Scholar
12.Watson, G. N.. A Treatise on the Theory of Bessel Functions (Cambridge University Press, 1984).Google Scholar