a1 School of Mathematics, University Walk, Bristol, BS8 1TW
If E is a subset of n (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 n 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.
(Received August 09 1984)