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 but it isn't really clear to me how. Any pointers (or even conjectures) here would be greatly appreciated!