Approximating reals by sums of rationals

Abstract

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 $1\le q_1,q_2,...,q_n\le N$. This leads to an inquiry on approximating a real number by rational numbers with a prescribed number of prime factors in the denominator.

8 pages