Approximating reals by sums of two rationals

Abstract

We generalize Dirichlet's diophantine approximation theorem to approximating any real number $\alpha$ by a sum of two rational numbers $\frac{a_1}{q_1}+\frac{a_2}{q_2}$ with denominators $1\le q_1,q_2\le N$. This turns out to be related to the congruence equation problem $xy\equiv c(\mathrm{mod}q)$ with $1\le x,y\le q^{1/2+\epsilon }$.

13 pages, improved results and some changes in the proofs

References in corpus (1)

• On a symmetric congruence and its applications