  We prove explicit versions of the Kronecker-Weyl theorems, both in a discrete
and a continuous settings, without any linear independence hypothesis. As an
application, we propose an alternative approach to problems concerning
asymptotic densities in prime number races, over number fields and over
function fields in one variable over finite fields, in the language of random
variables. Our approach allows us to prove new results on the existence and
positivity of some of those densities, which, in the case of races over
function fields, do not require any linear independence hypothesis.
