Let $A \in \mathbb{R}^{d\times d}$ be an invertible matrix. Consider the set $$P_d := A\mathbb{Z}^d = \{A x| x \in \mathbb{Z}^d \} \subset \mathbb{R}^d$$. and $$ Q_d := [-1,1]^d.$$
I am interest in enumerating (not just counting) all the points in $$Q_d \cap P_d.$$
Unfortunately I am not familiar with discrete mathematics/optimization and related subjects. Hence I would appreciate pointers to established methods/algorithms and literature, that are suited to tackle this problem.
Thank you.
Let $B = \left({A\atop -A}\right)$ be a matrix, whose rows are formed by the rows of matrices $A$ and $-A$. Then $$C = \{ x\in\mathbb{R}^d \mid Bx \leq u \},$$ where $u=(\underbrace{1,1,\dots,1}_{2d})^T$, is a convex polyhedron. Furthermore, $$Q_d \cap P_d = \{ Ax \mid x\in C\cap \mathbb{Z}^d \}$$ and thus the problem is reduced to enumerating the integral points in the polyhedron $C$.
There is a readily available software called LattE for lattice point enumeration, which can find the integral points of $C$ in the form of multivariate generating function (with the command "count --multivariate-generating-function"). While it requires elements of the matrix be integer, it is not hard to achieve that by taking an appropriate rational approximations and multiplying $Bx\leq u$ by an integer constant to make all entries integer.
See LattE documentation for further references and examples.
