Discrete tomography determination of bounded lattice sets from four X-rays

We deal with the question of uniqueness, namely to decide when an unknown finite set of points in $\mathbb{Z}^2$ is uniquely determined by its $X$-rays corresponding to a given set $S$ of lattice directions. In \cite{Ha} L. Hajdu proved that for any fixed rectangle $\mathcal{A}$ in $\mathbb{Z}^2$ there exists a valid set $S$ of four lattice directions (at least when $\mathcal{A}$ is not too ``small''), depending only on the size of $\mathcal{A}$, such that any two subsets of $\mathcal{A}$ can be distinguished by means of their $X$-rays taken in the directions in $S$. The proof was given by explicitly constructing a suitable set $S$ in any possible case. We improve this result by showing that in fact, for any fixed rectangle $\mathcal{A}$ in $\mathbb{Z}^2$, whole families of suitable sets of four directions can be found, for which we provide a complete characterization. Moreover this characterization permits to easily solve some relevant related problems.