Abstract
We give an integrability condition on a function psi guaranteeing that for almost all (or almost no) x is an element of R, the system vertical bar qx - p vertical bar < psi(t), vertical bar q vertical bar < t is solvable in p is an element of Z, q is an element of Z backslash {0} for sufficiently large t. Along the way, we characterize such x in terms of the growth of their continued fraction entries, and we establish that Dirichlet's Approximation Theorem is sharp in a very strong sense. Higher-dimensional generalizations are discussed at the end of the paper.