  There has been great interest in developing a theory of "Khintchine types"
for manifolds embedded in Euclidean space, and considerable progress has been
made for curved manifolds. We treat the case of translates of coordinate
hyperplanes, decidedly flat manifolds. In our main results, we fix the value of
one coordinate in Euclidean space and describe the set of points in the fiber
over that fixed coordinate that are rationally approximable at a given rate. We
identify translated coordinate hyperplanes for which there is a dichotomy as in
Khintchine's Theorem: the set of rationally approximable points is null or
full, according to the convergence or divergence of the series associated to
the desired rate of approximation.
