Non-additive bounded sets of uniqueness in $Z^n$

A main problem in discrete tomography consists in looking for theoretical models which ensure uniqueness of reconstruction. To this, lattice sets of points, contained in a multidimensional grid $\mathcal{A}=[m_1]\times [m_2]\times\dots \times [m_n]$ (where for $p\in\mathbb{N}$, $[p]=\{0,1,...,p-1\}$), are investigated by means of $X$-rays in a given set $S$ of lattice directions. Without introducing any noise effect, one aims in finding the minimal cardinality of $S$ which guarantees solution to the uniqueness problem. In a previous work the matter has been completely settled in dimension two, and later extended to higher dimension. It turns out that $d+1$ represents the minimal number of directions one needs in $\mathbb{Z}^n$ ($n\geq d\geq 3$), under the requirement that such directions span a $d$-dimensional subspace of $\mathbb{Z}^n$. Also, those sets of $d+1$ directions have been explicitly characterized. However, in view of applications, it might be quite difficult to decide whether the uniqueness problem has a solution, when $X$-rays are taken in a set of more than two lattice directions. In order to get computational simpler approaches, some prior knowledge is usually required on the object to be reconstructed. A powerful information is provided by additivity, since additive sets are reconstructible in polynomial time by using linear programming. In this paper we compute the proportion of non-additive sets of uniqueness with respect to additive sets in a given grid $\mathcal{A}\subset \mathbb{Z}^n$, in the important case when $d$ coordinate directions are employed.