USING ALMOST-EVERYWHERE THEOREMS FROM ANALYSIS TO STUDY RANDOMNESS
Kenshi Miyabe, André Nies, Jing Zhang · Bulletin of Symbolic Logic · 2016
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–Löf (ML) randomness. We establish several equivalences. Given a ML-random realz, the additional randomness strengths needed for the following are equivalent. (1) all effectively closed classes containingzhave density 1 atz. (2) all nondecreasing functions with uniformly left-c.e. increments are differentiable atz. (3) zis 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.