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], [7] and [8] to deal with bounded S   non-additive sets of uniqueness, where S⊂ZnS⊂Zn contains d   coordinate directions {e1,…,ed}{e1,…,ed}, |S|=d+1|S|=d+1, and n≥d≥3n≥d≥3. We prove that, when the union of two special subsets of {e1,…,ed}{e1,…,ed} has cardinality k=nk=n, then bounded S   non-additive sets of uniqueness are confined in a grid AA having a suitable fixed size in each coordinate direction eiei, whereas, if kki>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.