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

[2412.08997] Homometric subsets of $\mathbb{Z}_n$ with cardinality 5: classification and enumeration Skip to main content Search Submit Donate Log in Mathematics > Combinatorics arXiv:2412.08997 (math) [Submitted on 12 Dec 2024 ( v1 ), last revised 27 Aug 2026 (this version, v4)] Title: Homometric subsets of $\mathbb{Z}_n$ with cardinality 5: classification and enumeration Authors: William Q. Erickson , Nicholas B. Jones View PDF HTML (experimental) Abstract: 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 $n$. The same problem for $k \leq 4$ was partially solved by Erdős and ultimately settled by Rosenblatt-Berman (1984). 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. Comments: Updated Section 5 to reflect the recent solution of some open problems therein Subjects: Combinatorics (math.CO) MSC classes: 94A12 (Primary) 05A15, 00A65 (Secondary) Cite as: arXiv:2412.08997 [math.CO] (or arXiv:2412.08997v4 [math.CO] for this version) https://doi.org/10.48550/arXiv.2412.08997 Focus to learn more arXiv-issued DOI via DataCite Submission history From: William Erickson [ view email ] [v1] Thu, 12 Dec 2024 06:55:16 UTC (19 KB) [v2] Fri, 13 Jun 2025 04:54:30 UTC (37 KB) [v3] Fri, 7 Aug 2026 22:08:28 UTC (92 KB) [v4] Thu, 27 Aug 2026 18:17:46 UTC (92 KB) Full-text links: Access Paper: View PDF HTML (experimental) TeX Source view license Current browse context: math.CO < prev | next > new | recent | 2024-12 Change to browse by: math References & Citations SciX Google Scholar Semantic Scholar export BibTeX citation Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer ( What is the Explorer? ) Connected Papers Toggle Connected Papers ( What is Connected Papers? ) Litmaps Toggle Litmaps ( What is Litmaps? ) scite.ai Toggle scite Smart Citations ( What are Smart Citations? ) Code, Data, Media Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv ( What is alphaXiv? ) Links to Code Toggle CatalyzeX Code Finder for Papers ( What is CatalyzeX? ) DagsHub Toggle DagsHub ( What is DagsHub? ) GotitPub Toggle Gotit.pub ( What is GotitPub? ) Huggingface Toggle Hugging Face ( What is Huggingface? ) ScienceCast Toggle ScienceCast ( What is ScienceCast? ) Demos Demos Replicate Toggle Replicate ( What is Replicate? ) Spaces Toggle Hugging Face Spaces ( What is Spaces? ) Spaces Toggle TXYZ.AI ( What is TXYZ.AI? ) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower ( What are Influence Flowers? ) Core recommender toggle CORE Recommender ( What is CORE? ) Author Venue Institution Topic About arXivLabs arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs . Which authors of this paper are endorsers? | Disable MathJax ( What is MathJax? ) We gratefully acknowledge support from our major funders , member institutions , and all contributors. About · Help · Contact · Subscribe · Copyright · Privacy · Accessibility · Operational Status (opens in new tab)

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