Mathematical Proceedings of the Cambridge Philosophical Society

Research Article

Packing-dimension profiles and fractional Brownian motion

DAVAR KHOSHNEVISANa1 and YIMIN XIAOa2

a1 Department of Mathematics, 155 S. 1400 E., JWB 233, University of Utah, Salt Lake City, UT 84112–0090, U.S.A. e-mail: davar@math.utah.edu URL: http://www.math.utah.edu/~davar

a2 Department of Statistics and Probability, A-413 Wells Hall, Michigan State University, East Lansing, MI 48824, U.S.A. e-mail: xiao@stt.msu.edu URL: http://www.stt.msu.edu/~xiaoyimi

Abstract

In order to compute the packing dimension of orthogonal projections Falconer and Howroyd [3] have introduced a family of packing dimension profiles Dims that are parametrized by real numbers s > 0. Subsequently, Howroyd [5] introduced alternate s-dimensional packing dimension profiles P-Dims by using Caratheodory-type packing measures, and proved, among many other things, that P-Dims E = Dims E for all integers s > 0 and all analytic sets E xs2286 RN.

The aim of this paper is to prove that P-Dims E = Dims E for all real numbers s > 0 and analytic sets E xs2286 RN. This answers a question of Howroyd [5, p. 159]. Our proof hinges on establishing a new property of fractional Brownian motion.

(Received November 13 2006)

(Revised October 23 2007)

Footnotes

† Research partially supported by NSF grant DMS-0404729.