Denjoy, Demuth, and Density

We consider effective versions of two classical theorems, the Lebesgue density theorem and the Denjoy-Young-Saks theorem. For the first, we show that a Martin-Löf random real z ∈ [0, 1] is Turing incomplete if and only if every effectively closed class C ⊆ [0, 1] containing z has positive density at z. Under the stronger assumption that z is not LR-hard, we show that every such class has density one at z. These results have since been applied to solve two open problems on the interaction between the Turing degrees of Martin-Löf random reals and K-trivial sets: the non-cupping and covering problems. We say that f : [0, 1] → R satisfies the Denjoy alternative at z ∈ [0, 1] if either the derivative f ′(z) exists, or the upper and lower derivatives at z are +∞ and −∞, respectively. The Denjoy-Young-Saks theorem states that every function f : [0, 1]→ R satisfies the Denjoy alternative at almost every z ∈ [0, 1]. We answer a question posed by Kučera in 2004 by showing that a real z is computably random if and only if every computable function f satisfies the Denjoy alternative at z. For Markov computable functions, which are only defined on computable reals, we can formulate the Denjoy alternative using pseudo-derivatives. Call a real z DA-random if every Markov computable function satisfies the Denjoy alternative at z. We considerably strengthen a result of Demuth (Comment. Math. Univ. Carolin., 24(3):391–406, 1983) by showing that every Turing incomplete Martin-Löf random real is DA-random. The proof involves the notion of non-porosity, a variant of density, which is the bridge between the two themes of this paper. We finish by showing that DA-randomness is incomparable with Martin-Löf randomness.