  We solve the convergence case of the generalized Baker-Schmidt problem for
simultaneous approximation on affine subspaces, under natural diophantine type
conditions. In one of our theorems, we do not require monotonicity on the
approximation function. In order to prove these results, we establish
asymptotic formulae for the number of rational points close to an affine
subspace. One key ingredient is a sharp upper bound on a certain sum of
reciprocals of fractional parts associated with the matrix defining the affine
subspace.
