### Using Almost-Everywhere Theorems from Analysis to Study Randomness

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

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.

### Randomness notions in Muchnik and Medvedev degrees

News
20 Sep, 2016, the slide file was uploaded

Title
Randomness notions in Muchnik and Medvedev degrees

Type
Talk at a meeting of CTFM2016

abstract
slide

### Computability of reals on the space where the triangle inequality does not hold

News
2 Sep, 2016, the slide file was uploaded

Title
Computability of reals on the space where the triangle inequality does not hold

Type
Talk at a meeting of Mathematical Society of Japan

slide

### Computability of reals on the space where the triangle inequality does not hold

News
25 July, 2016, the slide file was uploaded

Title
Computability of reals on the space where the triangle inequality does not hold

Type
Talk at a lab seminar