Homometric subsets of $\mathbb{Z}_n$ with cardinality 5: classification and enumeration

Two subsets of $\mathbb{Z}_n$ are said to be homometric if they have the same multiset of pairwise cyclic (i.e., Lee) distances. Homometric subsets necessarily have the same cardinality, say $k$. In this paper, for all positive integers $n$, we classify the homometric subsets of $\mathbb{Z}_n$ with cardinality $k=5$ (modulo cyclic shifts and reflections). Our classification consists of six families of homometric pairs, and one family of homometric triples. We also give a closed-form generating function that counts these homometric pairs and triples for all $n$. As an immediate application of our result, one obtains an explicit criterion for the solvability of the crystallographic phase retrieval problem, in the setting of binary signals supported on $k=5$ many atoms. The same problem for $k \leq 4$ was partially solved by Erd\H{o}s and ultimately settled by Rosenblatt-Berman (1984), who noted that for $k \geq 5$ the problem seems very difficult. Equivalently, in the language of microtonal music theory, our result solves the open problem of classifying Z-related pentachords.