We obtain analogues of two famous classical Hermann Weyl’s theorems on the uniform distribution of the sequence of fractional parts of polynomial and exponential functions for positive integer arguments. These results can be obtained in the case of the change of the condition "almost everywhere with respect to Lebesgue measure" to the condition "almost everywhere in the Baire sense" (as well known corresponding properties are not follows from each other), as well as, by the change of the condition "uniform distribution" to the condition "everywhere dense".