Counting number of points on a lattice in a hypercube

Question

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!

Answer 1

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, 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.