Let $A:\mathbb{Z}^n\to \mathbb{Z}^n$ be non-singular. Consider a box $B=[0,N_1]\times [0,N_2] \times \dotsc \times [0,N_n]$. Let $p_1,\dotsc,p_n$ be primes (distinct, if you wish) and let $L = p_1\mathbb{Z} \times \dotsb \times p_n \mathbb{Z}$. Can one give a good bound on $A^{-1} L \cap B$? For instance, might $$|A^{-1} L \cap B| \ll \det(A) \prod_{i\leq n} (N_i/p_{\pi(i)} + 1)$$ hold for some permutation $\pi$ of $\{1,2,\dotsc,n\}$?

Assume all $p_i$ considerably larger than $n$ if needed. Also assume the matrix entries of $A$ to be bounded, if needed.

Note: I am not looking for an estimate with error term of size $\prod_{j\leq n-1} N_j$, say. That would be easy.