Hausdorff dimension and oracle constructions

Bennett and Gill (1981) proved that P 6= NP relative to a random oracle A, or in other words, that the set O[P=NP] = {A | P = NP} 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 Φ, then the set of oracles relative to which Φ 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 6= NP ∩ coNP relative to a Hausdorff dimension 1 set of oracles.