Kronecker's Approximation Theorem

A theorem characterizing when linear combinations of real numbers with integer coefficients are dense modulo 1 in a torus: if 1, α1,...,αn are linearly independent over Q then the set of vectors (mα1 mod 1, ..., mαn mod 1) for m∈Z is dense in the n-torus; more generally it describes density of the subgroup generated by given real vectors modulo 1 under rational independence conditions.