Inverse characterization of critical pairs for restricted sumsets

Characterize all pairs of nonempty subsets A,B of the cyclic group Z_p with |A|≠|B| that attain equality in the Alon–Nathanson–Ruzsa bound |A+B|=|A|+|B|−2, thereby solving the unresolved inverse problem for restricted sumsets in Z_p.

Background

For nonempty subsets A,B⊂Z_p with unequal cardinalities, the Alon–Nathanson–Ruzsa theorem gives the lower bound |A+B|≥min{p,|A|+|B|−2} for the restricted sumset, in which equal summands are excluded. The corresponding inverse problem asks for a complete description of all critical pairs attaining equality. The paper notes that this problem was posed by Alon, Nathanson, and Ruzsa and remains unresolved in general. It further explains that the asymmetric case is still largely open, while the paper addresses a particular conjectural classification under the additional restriction |A|+|B|≤p−1.