Let $\alpha_1, \alpha_2, \dots$ be an infinite sequence of real numbers such that any finite subset is linearly independent over $\mathbb{Q}$. Let $f(N)$ be the number of tuples $(m_1, \dots, m_N)$ with $m_i \in \mathbb{Z}$, $|m_i| \leq N$.
 for which
$$\left|\sum_{i=1}^{N} m_i \alpha_i \right|< \frac{1}{N^{100}}.$$

(where 100 could be your favorite constant). Is there any known way to put an asymptotic upper bound on $f(N)$? 

For concreteness (or if it makes the question possible to answer!) you could take the $\alpha_i$ to be the square roots of square free integers (ordered however you like, although I do have a particular ordering in mind). If that doesn't work, any effectively computable $\alpha_i$ would be welcome.

It seems that one *might* be able to say something of the sort using [Evertse's quantitative subspace theorem](https://arxiv.org/abs/1008.2268) but it isn't really clear to me how. Any pointers (or even conjectures) here would be greatly appreciated!