  We prove a version of the Khinchine--Groshev theorem for Diophantine
approximation of matrices subject to a congruence condition. The proof relies
on an extension of the Dani correspondence to the quotient by a congruence
subgroup. This correspondence together with a multiple ergodic theorem are used
to study rational approximations in several congruence classes simultaneously.
The result in this part holds in the generality of weighted approximation but
is restricted to simple approximation functions.
