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. 



Number of solutions of linear homogenous Diophantine equation inside a box
The solution set $L$ is clearly a sublattice in $\mathbb Z^d$. If $a$ is fixed and $N$ grows, the number of lattice elements in a box grows approximately as $cN^{rank}$. So if you have your statement asymptotically for $N\to+\infty$ for a fixed $a$ then this shows that the rank of the solutions set is $d-1$, so $a$ is rational (see also Richard Stanley's answer). 


To bound the size of $a$ consider the following. Suppose for simplicity that all $N\_i$ are equal to $N$ and that the number of solutions is $> cN^{d-1}$. Then for $N\_0=[2^{d-2}/c]+1$
the number of solutions is larger than the number of points in $(-N,N)^{d-2}$. This implies
that for any $i,j=1,...,d$ there are at least two solutions of size less than $N\_0$ which have the same projection to the $\mathbb Z^{d-2}$ obtained by forgetting $i$-th and $j$-th coordinates. Thus, there is a nonzero element in the lattice $L$ with nonzero coordinates at $l\_i$ and $l\_j$ only, and $\vert l\_i\vert,\vert l\_j\vert < 2N\_0$. We may assume that $a\_i$ are all nonzero, then this gives us a finite set of possibilities of their ratios, hence a finite set of $(a\_i)$ up to common scaling. 


