Abstract
We give an integrability criterion on a real-valued non-increasing function t/) guaranteeing that for almost all (or almost no) pairs (A, b), where A is a real rn x n matrix and b Ikm, the system
parallel to Aq+b - P parallel to(m) < psi(T), parallel to q parallel to(n) < T,
is solvable in p E Zrn, q E Zn for all sufficiently large T. The proof consists of a reduction to a shrinking target problem on the space of grids in rel+Th. We also comment on the homogeneous counterpart to this problem, whose rn = n = 1 case was recently solved, but whose general case remains open.