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
