Bennett and Gill (1981) proved that P^A != NP^A relative to a
random oracle A, or in other words, that the set
O_[P=NP] = { A | 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 != NP^A intersect coNP^A
relative to a Hausdorff dimension 1 set of oracles.