# Jeremy Avigad: Uniform distribution and algorithmic randomness II

Mathematical logic seminar – October 28, 2014
Time: 12:30 – 13:30

Room: Wean Hall 8220

Department of Philosophy
CMU

Title: Uniform distribution and algorithmic randomness II

Abstract:

A seminal theorem due to Weyl states that if (a_n) is any sequence of distinct
integers, then, for almost every real number x, the sequence (a_n x) is
uniformly distributed modulo one. In particular, for almost every x in the unit
interval, the sequence (a_n x) is uniformly distributed modulo one for every
*computable* sequence (a_n) of distinct integers. Call such an x UD random.

Every Schnorr random real is UD random, but there are Kurtz random reals that
are not UD random. On the other hand, Weyl’s theorem still holds relative to a
particular effectively closed null set, so there are UD random reals that are
not Kurtz random.

In these talks, I will prove Weyl’s theorem and provide the relevant background
from algorithmetic randomness, and then discuss the results above.