I have a set $\{a_i \in \Bbb R | \ i <=7 \}$, and I'm looking for a way to find values of $x$ where given $\epsilon > 0$, $$\forall i \ \exists n_i \in \Bbb{Z} \ \ |a_i x - n_i| < \epsilon$$
Does anyone know a constructive algorithm to find values of $x$ where this would hold?
Edit: $x \neq 0$ and $\forall i \ n_i \neq 0$.
Edit 2: I realized that I haven't really showed my work. I'm not that familiar with this side of mathematics so my attempts are probably naive, but here it is. My attempt to approximate $$f(x) = \sum_{i=0}^7 | g(a_i x) - a_i x| $$ (where $g(y)$ rounds $y$ to the nearest integer) with a polynomial seemed dumb as $f$ has at least a million turning points in its domain.
I looked at the simultaneous version of Dirichlet's approximation theorem to get me at least to rational numbers, but I couldn't find a constructive proof that seems to work for 7 numbers.
From what I've looked at the LLL algorithm, it looks like it does something very close to what I want, but I don't understand how to apply it in this specific case, where I don't have a reasonable polynomial approximation, and I'm not looking for an integer relation.