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 S non-additive sets of uniqueness, where S⊂Zn contains d coordinate directions {e_1,.., e_d}, |S|=d+1, and n≥d≥3. We prove that, when the union of two special subsets of {e_1,.., e_d} has cardinality k=n, then bounded S non-additive sets of uniqueness are confined in a grid A having a suitable fixed size in each coordinate direction ei, whereas, if k<n, the grid A can be arbitrarily large in each coordinate direction ei, where i>k. The subclass of pure bounded S non-additive sets plays a special role. We also compute explicitly the proportion of bounded S non-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.