Polynomial-Time Rademacher Theorem, Porosity and Randomness

The main result of this paper is a polynomial time version of Rademacher’s theorem. We show that if z ∈ R is p-random, then every polynomial time computable Lipschitz function f : R → R is differentiable at z. This is a generalization of the main result of [19]. To prove our main result, we introduce and study a new notion, p-porosity, and prove several results of independent interest. In particular, we characterize p-porosity in terms of polynomial time computable martingales and we show that p-randomness in R is invariant under polynomial time computable linear isometries. 1998 ACM Subject Classification F.1.1 Models of Computation