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 $f(N)$?

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!