Hausdorff dimension and oracle constructions

Bennett and Gill [Relative to a random oracle A, p(A) not equal Np-A not equal co-Np-A with probability 1, SIAM J. Comput. 10 (1981) 96-113] proved that p(A) not equal Np-A relative to a random oracle A, or in other words, that the set O-[P=NP] = {A vertical bar p(A) = Np-A} has Lebesgue measure 0. In contrast, we show that O-[P=NP] has Hausdorff dimension 1. This follows from a much more general theorem: if there is a relativizable and paddable oracle construction for a complexity-theoretic statement Phi, then the set of oracles relative to which Phi holds has Hausdorff dimension 1. We give several other applications including proofs that the polynomial-time hierarchy is infinite relative to a Hausdorff dimension 1 set of oracles and that p(A) not equal Np-A boolean AND coNp(A) relative to a Hausdorff dimension I set of oracles. (c) 2006 Elsevier B.V. All rights reserved.