Mathematical Proceedings of the Cambridge Philosophical Society

Research Article

Packing dimensions of projections and dimension profiles

K. J. FALCONER a1 and J. D. HOWROYD a1
a1 Mathematical Institute, University of St Andrews, North Haugh, St Andrews, Fife, KY16 9SS, Scotland


For E a subset of [open face R]n and 0 [less-than-or-eq, slant] m [less-than-or-eq, slant] n we define a ‘family of dimensions’ DimmE, closely related to the packing dimension of E, with the property that the orthogonal projection of E onto almost all m-dimensional subspaces has packing dimension DimmE. In particular the packing dimension of almost all such projections must be equal. We obtain similar results for the packing dimension of the projections of measures. We are led to think of DimmE for m [set membership] [0, n] as a ‘dimension profile’ that reflects a variety of geometrical properties of E, and we characterize the dimension profiles that are obtainable in this way.

(Received July 6 1995)
(Revised October 3 1995)