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| Natura: | Preprint |
| Pubblicazione: |
2024
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| Accesso online: | https://arxiv.org/abs/2412.08997 |
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| _version_ | 1866918057231777792 |
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| author | Erickson, William Q. Jones, Nicholas B. |
| author_facet | Erickson, William Q. Jones, Nicholas B. |
| contents | 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 Erdős 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_08997 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Homometric subsets of $\mathbb{Z}_n$ with cardinality 5: classification and enumeration Erickson, William Q. Jones, Nicholas B. Combinatorics 94A12 (Primary) 05A15, 00A65 (Secondary) 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 Erdős 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. |
| title | Homometric subsets of $\mathbb{Z}_n$ with cardinality 5: classification and enumeration |
| topic | Combinatorics 94A12 (Primary) 05A15, 00A65 (Secondary) |
| url | https://arxiv.org/abs/2412.08997 |