Denjoy, Demuth, and Density

TL;DR

This paper explores effective versions of classical theorems related to randomness and density, establishing new connections between Turing degrees, randomness notions, and differentiability properties of functions.

Contribution

It introduces effective density and Denjoy-Young-Saks theorems, linking Martin-Loef randomness with computable functions and strengthening previous results on randomness and density.

Findings

Martin-Loef random reals are Turing incomplete iff they are in classes with positive density.

A real is computably random iff all computable functions satisfy the Denjoy alternative at it.

Every Turing incomplete Martin-Loef random real is DA-random, strengthening Demuth's result.

Abstract

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-Loef random real $z\in [0,1]$ is Turing incomplete if and only if every effectively closed class $C \subseteq [0,1]$ containing $z$ has positive density at $z$. Under the stronger assumption that $z$ is not LR-hard, we show that $z$ has density-one in every such class. These results have since been applied to solve two open problems on the interaction between the Turing degrees of Martin-Loef random reals and $K$-trivial sets: the non-cupping and covering problems. We say that $f\colon[0,1]\to\mathbb{R}$ satisfies the Denjoy alternative at $z \in [0,1]$ if either the derivative $f'(z)$ exists, or the upper and lower derivatives at $z$ are $+\infty$ and $-\infty$, respectively. The Denjoy-Young-Saks theorem states that every function $f\colon[0,1]\to\mathbb{R}$ satisfies the Denjoy alternative at almost every $z\in[0,1]$. We answer a question posed by Kucera 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-Loef 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-Loef randomness.