Details

Autor(en) / Beteiligte

• MIYABE, KENSHI
• NIES, ANDRÉ
• ZHANG, JING

Titel

USING ALMOST-EVERYWHERE THEOREMS FROM ANALYSIS TO STUDY RANDOMNESS

Ist Teil von

• The bulletin of symbolic logic, 2016-09, Vol.22 (3), p.305-331

Ort / Verlag

New York, USA: Cambridge University Press

Erscheinungsjahr

2016

Quelle

Alma/SFX Local Collection

Beschreibungen/Notizen

• 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–Löf (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 related to density of ${\rm{\Pi }}_n^0$ and ${\rm{\Sigma }}_1^1$ classes at a real.