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On the Hausdorff dimensions of distance sets

  • K. J. Falconer (a1)
  • DOI: http://dx.doi.org/10.1112/S0025579300010998
  • Published online: 01 February 2010
Abstract

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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