Uniform distribution and algorithmic randomness

Details Uniform distribution and algorithmic randomness Uniform distribution and algorithmic randomness

2012-03-19

arxiv.org/abs/1203.4126v1

Mathematics

Logic

Scientific paper

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". 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.

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