Approximating reals by sums of rationals

We consider the question of approximating any real number a by sums of n rational numbers a(1)/q(1) + a(2)/q(2) + ... + a(n)/q(n) with denominators 1 <= q(1), q(2), ..., q(n) <= N. This leads to inquiries on approximating a real number by rational numbers with a prescribed number of prime factors in the denominator as well as by rational numbers with smooth denominator. (C) 2008 Elsevier Inc. All rights reserved.