Research Article

On the Hausdorff dimensions of distance sets

K. J. Falconera1

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 [11], 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)

Key Words:

  • 28A75: MEASURE AND INTEGRATION; Classical Measure theory; Geometric Measure theory