Discrete Tomography determination of bounded sets in $Z^n$.

This paper deals with the uniqueness problem for bounded sets in Discrete Tomography, namely to decide when an unknown finite subset of points in a multidimensional grid is uniquely determined by its X-rays corresponding to a given set of lattice directions. In Brunetti et al. (2013) the authors completely settled the problem in dimension two, by proving that whole families of suitable sets of four directions can be found, for which they provided a complete characterization. In this paper we examine the problem in higher dimensions. We show that d + 1 represents the minimal number of directions we need in Zn (n ≥ d ≥ 3) to distinguish all the subsets of a finite grid by their X-rays in these directions, under the requirement that such directions span a d-dimensional subspace of Zn. It turns out that the situation differs from the planar case, where less than four directions are never sufficient for uniqueness. Moreover we characterize the minimal sets of directions of uniqueness by providing a necessary and sufficient condition. As a consequence, results in Brunetti et al. (2013) can be easily extended to the n-dimensional case