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We prove explicit versions of the Kronecker-Weyl theorems, both in a discreteand a continuous settings, without any linear independence hypothesis. As anapplication, we propose an alternative approach to problems concerningasymptotic densities in prime number races, over number fields and overfunction fields in one variable over finite fields, in the language of randomvariables. Our approach allows us to prove new results on the existence andpositivity of some of those densities, which, in the case of races overfunction fields, do not require any linear independence hypothesis.

Explicit Kronecker-Weyl theorems and applications to prime number races