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BY A. F. BEARDON Communicated by L. Bers, March 26, 1965 1. Hausdorff dimension. Suppose E is a compact subset of Ndimensional euclidean space, E. We denote by ma(E) the Hausdorff a-dimensional measure of E and by d(E) the Hausdorff dimension of E% i.e. the unique non-negative number such that ma(E) = 0 for a > d(E) and ma(E) = + oo for 0 ^ a < d{E). We shall need the following result. THEOREM A [6]. Let E be a compact subset of E. Then d(E)>0 implies E has positive logarithmic capacity. 2. Spherical Cantor sets. DEFINITION 1 [2], [7]. We say E is a spherical Cantor set if and only if E can be expressed in the form

The Hausdorff dimension of singular sets of properly discontinuous groups in 𝑁-dimensional space