Mathematical logic seminar - October 14, 2014

Time: 12:30 - 13:30

Room: Wean Hall 8220

Speaker:     Jeremy Avigad    
Department of Philosophy

Title: Uniform distribution and algorithmic randomness I


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