Randomness and Uniform Distribution

SLAMAN, THEODORE A.; BECHER, VERÓNICA; GRIGOREFF, SERGE

Oberwolfach

Workshop; Oberwolfach Workshop 1802 "Computability Theory''; 2018

The Mathematisches Forschungsinstitut Oberwolfach (MFO, Oberwolfach Research Institute for Mathematics)

Abstract: How is the notion of randomness of algorithmic information theory related tothe notion of uniform distribution? In this talk we study Martin-Löf randomness for real numbers in terms of uniform distribution of sequences. We present a necessary conditionfor a real number to be Martin-Löf random, and a strengthening of that conditionwhich is sufficient for Martin-Löf randomness. For this strengthening we define a notionof uniform distribution relative to the computably enumerable open subsets of the unit interval. We call the notion Sigma^0_1-uniform distribution.