On bounded additivity in discrete tomography

In this paper we investigate bounded additivity in Discrete Tomography. This notion has been previously introduced in [5], as a generalization of the original one in [11], which was given in terms of ridge functions. We exploit results from [6–8]to deal with bounded Snon-additive sets of uniqueness, where S⊂Zncontains dcoordinate directions {e1, ..., ed}, |S| =d +1, and n ≥d ≥3. We prove that, when the union of two special subsets of {e1, ..., ed}has cardinality k =n, then bounded Snon-additive sets of uniqueness are confined in a grid Ahaving a suitable fixed size in each coordinate direction ei, whereas, if k k. The subclass of pure bounded Snon-additive sets plays a special role. We also compute explicitly the proportion of bounded Snon-additive sets of uniqueness w.r.t. those additive, as well as w.r.t. the S-unique sets. This confirms a conjecture proposed by Fishburn et al. in [14]for the class of bounded sets.