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 , 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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