Using almost-everywhere theorems from analysis to study randomness

TL;DR

This paper explores new notions of algorithmic randomness derived from effective versions of classical theorems in analysis and ergodic theory, establishing equivalences and examining their implications for randomness and density properties.

Contribution

It introduces stronger randomness notions based on effective almost-everywhere theorems and proves their equivalences, extending the understanding of randomness in computability theory.

Findings

Equivalence of randomness notions related to density and differentiability at ML-random reals.

Characterization of randomness via convergence of semicomputable martingales.

Analysis of randomness for classes defined by higher-level descriptive set theory.

Abstract

We study algorithmic randomness notions via effective versions of almost-everywhere theorems from analysis and ergodic theory. The effectivization is in terms of objects described by a computably enumerable set, such as lower semicomputable functions. The corresponding randomness notions are slightly stronger than \ML\ (ML) randomness. We establish several equivalences. Given a ML-random real $z$, the additional randomness strengths needed for the following are equivalent. \n (1) all effectively closed classes containing $z$ have density $1$ at $z$. \n (2) all nondecreasing functions with uniformly left-c.e.\ increments are differentiable at $z$. \n (3) $z$ is a Lebesgue point of each lower semicomputable integrable function. We also consider convergence of left-c.e.\ martingales, and convergence in the sense of Birkhoff's pointwise ergodic theorem. Lastly we study randomness notions for density of $\Pi^0_n$ and $\Sigma^1_1$ classes.