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. 