Suppose I have a lattice $\Lambda \in \mathbb{R}^n$. Let $X\_i >0$ for $i=1,..,n$. I am interested in some references regarding counting number of points of $\Lambda$ inside $[-X\_1, X\_1] \times \cdots \times [-X\_n, X\_n]$. Could someone point me to some references related to this. Thank you very much!



Counting number of points on a lattice in a hypercube
After applying a suitable invertible linear transformation on $\mathbb{R}^n$, the lattice $\Lambda$ becomes $\mathbb{Z}^n$, and the box $[-X\_1, X\_1] \times \cdots \times [-X\_n, X\_n]$ becomes a parallelotope $\mathcal{R}$. So the task becomes to estimate the cardinality of $\mathcal{R}\cap\mathbb{Z}^n$. 


Now, by the main theorem of Davenport's paper [On a principle of Lipschitz](https://pdfs.semanticscholar.org/4ea0/092ec987de593a24fd21dd435167921210e9.pdf), we have
$$ \bigl|\#(\mathcal{R}\cap\mathbb{Z}^n)-\mathrm{vol}(\mathcal{R})\bigr| \leq \sum\_{m=0}^{n-1} V\_m,$$
where $V\_m$ is the sum of the $m$-dimensional volumes of the projections of $\mathcal{R}$ on the various coordinate spaces obtained by equating any $n—m$ coordinates to zero, and $V\_0=1$ by convention.


