Research Article

Hausdorff dimension and the exceptional set of projections

K. J. Falconera1

a1 The School of Mathematics, University of Bristol, University Walk, Bristol. BS8 1TW

If П is a k-dimensional vector subspace of Rn and E is a subset of Rn, let projп(E) denote the orthogonal projection of E onto П. Marstrand [8] and Kaufman [6] have developed results on the Hausdorff dimension and measure of projп(E) in terms of the dimension of E, leading to the very general theory of Mattila [11]. In particular, Mattila shows that if the Hausdorff dimension dim E of the Souslin set E is greater than k, then projп(E) has positive k-dimensional Lebesgue measure for almost all П xs2208 Gn, k (in the sense of the usual normalized invariant measure on the Grassmann manifold Gn, k of k-dimensional subspaces of Rn).

(Received April 01 1981)

Key Words:

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