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Approximating reals by sums of two rationals

Tsz Ho Chan
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Abstract

We generalize Dirichlet's diophantine approximation theorem to approximating any real number $α$ 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 $1 \leq x, y \leq q^{1/2 + ε}$.

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Tsz Ho Chan. 2007-04-20. Approximating reals by sums of two rationals. https://arxiv.org/abs/math/0609322

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