Uniform distribution and algorithmic randomness
Jeremy D. Avigad · Journal of Symbolic Logic · 2013
Abstract A seminal theorem due to Weyl [14] states that if (an) is any sequence of distinct integers, then, for almost everyx∈ ℝ, the sequence (anx) is uniformly distributed modulo one. In particular, for almost everyxin the unit interval, the sequence (anx) is uniformly distributed modulo one for everycomputablesequence (an) of distinct integers. Call such anx UD random. Here it is shown that 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.