Approximating reals by sums of two rationals

Details Approximating reals by sums of two rationals Approximating reals by sums of two rationals

2006-09-12

arxiv.org/abs/math/0609322v2

Mathematics

Number Theory

13 pages, improved results and some changes in the proofs

Scientific paper

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 $1 \leq q_1, q_2 \leq N$. This turns out to
be related to the congruence equation problem $x y \equiv c \pmod q$ with $1
\leq x, y \leq q^{1/2 + \epsilon}$.

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