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On the Hausdorff dimensions of distance sets

  • K. J. Falconer (a1)

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.

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1.Carleson, L.. Selected Problems on Exceptional Sets (Van Nostrand, Princeton, 1967).
2.Chung, F. R. K.. The number of different distances determined by n points in the plane J. Combinatorial Theory A, 26 (1984), 342354.
3.Davies, Roy O.. Subsets of finite measure in analytic sets. Indag. Math., 14 (1952), 488489.
4.Davies, Roy O.. Two counter-examples concerning Hausdorff dimensions of projections. Colloq. Math., 42 (1979), 5358.
5.Davies, Roy O.. Rings of dimension d. To appear.
6.Falconer, K. J.. The Geometry of Fractal Sets (Cambridge University Press, 1984).
7.Falconer, K. J.. Rings of fractional dimension. Mathematika, 31 (1984), 2527.
8.Falconer, K. J.. Classes of sets with large intersection. Mathematika, 32 (1985), 191205.
9.Marstrand, J. M.. The dimension of cartesian product sets. Proc. Cambridge Phil. Soc., 50 (1954), 198202.
10.Rogers, C. A.. Hausdorff Measures (Cambridge University Press, 1970).
11.Steinhaus, H.. Sur les distances des points des ensembles de mesure positive. Fund. Math., 1 (1920), 93104.
12.Watson, G. N.. A Treatise on the Theory of Bessel Functions (Cambridge University Press, 1984).
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  • ISSN: 0025-5793
  • EISSN: 2041-7942
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