Return to Revisions

For given $(a_1, a_2, a_3)$ and $(c_1, c_2 , c_3)$, the required condition is equivalent to: $(c_2-c_1, c_3-c_2)$ is in the closure $\Gamma $ of the additive subgroup generated by $(a_1,0)$, $(-a_2,a_2)$, $(0,-a_3)$. But the latter need not be dense, even if $(a_1, a_2, a_3)$ are linearly independent over the rationals; for instance $(a_1, a_2, a_3):=(1+\lambda,1,1+1/\lambda)$ with $\lambda$ an irrational non quadratic, is a linear independent triple for which $\Gamma$ is a family of lines $\{(x,y)\, : \, y=\lambda x + n(\lambda+1)\, , n\in\mathbb{N}\}$.