  We consider the question of approximating any real number $\alpha$ by sums of
$n$ rational numbers $\frac{a_1}{q_1} + \frac{a_2}{q_2} + ... +
\frac{a_n}{q_n}$ with denominators $1 \leq q_1, q_2, ..., q_n \leq N$. This
leads to an inquiry on approximating a real number by rational numbers with a
prescribed number of prime factors in the denominator.
