**News**

10 Oct 2016, published online

29 Feb 2016, Accepted by BSL

May 2015, Resubmitted.

3 Nov 2014. Submitted

**Title**

Using Almost-Everywhere Theorems from Analysis to Study Randomness

(with Jing Zhang and Andre Nies)

**Type**

Full paper

**Journal**

The Bulletin of Symbolic Logic, Volume 22, Issue 3

September 2016, pp. 305-331

arXiv

The latest version is here.

**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 Martin-Lo ̈f (ML) randomness. We establish several equivalences. Given a ML-random real z, the additional randomness strengths needed for the following are equivalent.

(1) all effectively closed classes containing z have density 1 at z.

(2) all nondecreasing functions with uniformly left-c.e. increments are differentiable at z.

(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.