On the Non-Additive Sets of Uniqueness in a Finite Grid

In Discrete Tomography there is a wide literature concerning (weakly) bad configurations. These occur in dealing with several questions concerning the important issues of uniqueness and additivity. Discrete lattice sets which are additive with respect to a given set $S$ of lattice directions are uniquely determined by $X$-rays in the direction of $S$. These sets are characterized by the absence of weakly bad configurations for $S$. On the other side, if a set has a bad configuration with respect to $S$, then it is not uniquely determined by the $X$-rays in the directions of $S$, and consequently it is also non-additive. Between these two opposite situations there are also the non-additive sets of uniqueness, which deserve interest in Discrete Tomography, since their unique reconstruction cannot be derived via the additivity property. In this paper we wish to investigate possible interplays among such notions in a given lattice grid $\mathcal{A}$, under $X$-rays taken in directions belonging to a set $S$ of four lattice directions.