  A reasonably complete theory of the approximation of an irrational by
rational fractions whose numerators and denominators lie in prescribed
arithmetic progressions is developed in this paper. Results are both, on the
one hand, from a metrical and a non-metrical point of view and, on the other
hand, from an asymptotic and also a uniform point of view. The principal
novelty is a Khintchine type theorem for uniform approximation in this context.
Some applications of this theory are also discussed.
