TITLE: Near points in several arithmetic progressions
QUESTION [11 upvotes]: We have 3 arithmetic progressions $na_1 + c_1, na_2 + c_2, na_3 + c_3$ with the reals $a_1,a_2,a_3$ linearly independent over the rationals. Is it always true that given $\epsilon > 0$ there are terms $x_1,x_2,x_3$, one from each progression, such that all three are within $\epsilon$ of each other ?
Same problem with any finite number of progressions.

REPLY [3 votes]: 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 $\{(x,y)\, : \,  y=\lambda x + n(\lambda+1)\, , n\in\mathbb{N}\}$.