TITLE: Number of solutions of linear homogenous Diophantine equation inside a box
QUESTION [9 upvotes]: Let $a_1, ..., a_d$ be positive reals and consider the linear Diophantine equation 
$$
\sum_i a_in_i = 0.
$$
I am interested in estimating the number of integer solutions of this equation inside a box
$[-N_1, N_1] \times ... \times [-N_d, N_d]$ and also the minimal basis for the corresponding integer lattice in terms of $a_1,...,a_d$. More specifically, it seems that if the number of solutions is of order $N^{d-1}$ then $(a_1, ...., a_d)$ should be proportional to some integer vector of bounded norm. Presumably, smth much more precise is known.
I believe that such kind of questions are well studied so any good reading on this topic would be highly appreciated.

REPLY [3 votes]: Siegel's Lemma is a famous solution to this type of problem with a system of linear forms, giving an estimate for the basis of solutions.