Jeremy Avigad: Uniform distribution and algorithmic randomness I

Mathematical logic seminar – October 14, 2014
Time:     12:30 – 13:30

Room:     Wean Hall 8220

Speaker:         Jeremy Avigad
Department of Philosophy
CMU

Title:     Uniform distribution and algorithmic randomness I

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.