Approximating reals by sums of two rationals

We generalize Dirichlet's diophantine approximation theorem to approximating any real number alpha by a sum of two rational numbers a(1)/q(1) + a(2)/q(2) with denominators 1 <= q(1), q(2) <= N. This turns out to be related to the congruence equation problem xy =c (mod q) with 1 <= x, y <= q(1/2+is an element of). (C) 2007 Elsevier Inc. All rights reserved.