Mathematical logic seminar – October 28, 2014

Time: 12:30 – 13:30

Room: Wean Hall 8220

Speaker: Jeremy Avigad

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.