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{
    "pk": 65000,
    "title": "On the determination of sets by their subset sums",
    "subtitle": null,
    "abstract": "Let \\(A\\) be a multiset with elements in an abelian group. Let \\(\\operatorname{FS}(A)\\) be the multiset containing the \\(2^{|A|}\\) sums of all subsets of \\(A\\). We study the reconstruction problem \"Given \\(\\operatorname{FS}(A)\\), is it possible to identify \\(A\\)?\". We prove that, up to identifying multisets through a natural equivalence relation, the function \\(A \\mapsto \\operatorname{FS}(A)\\) is injective (and thus the reconstruction problem is solvable) if and only if every order \\(n\\) of a torsion element of the abelian group satisfies a number-theoretical property related to the multiplicative group \\((\\mathbb{Z}/n \\mathbb{Z})^*\\). The core of the proof relies on a delicate study of the structure of cyclotomic units. Moreover, as a tool, we develop an inversion formula for a novel discrete Radon transform on finite abelian groups that might be of independent interest.\n \nMathematics Subject Classifications: 11P70, 05B10, 11R18, 44A12\n \nKeywords: Subset sums, inverse problems, Radon transform, cyclotomic extension",
    "language": "en",
    "license": {
        "name": "Creative Commons Attribution 4.0",
        "short_name": "CC BY 4.0",
        "text": "Attribution — You must give appropriate credit, provide a link to the license, and indicate if changes were made. You may do so in any reasonable manner, but not in any way that suggests the licensor endorses you or your use.\n\nNo additional restrictions — You may not apply legal terms or technological measures that legally restrict others from doing anything the license permits.",
        "url": "https://creativecommons.org/licenses/by/4.0"
    },
    "keywords": [
        {
            "word": "Subset sums"
        },
        {
            "word": "inverse problems"
        },
        {
            "word": "Radon transform"
        },
        {
            "word": "cyclotomic extension"
        }
    ],
    "section": "Research Articles",
    "is_remote": true,
    "remote_url": "https://escholarship.org/uc/item/9s01w2p2",
    "frozenauthors": [
        {
            "first_name": "Andrea",
            "middle_name": "",
            "last_name": "Ciprietti",
            "name_suffix": "",
            "institution": "Department of Mathematics, University of Pisa, Pisa, Italy",
            "department": ""
        },
        {
            "first_name": "Federico",
            "middle_name": "",
            "last_name": "Glaudo",
            "name_suffix": "",
            "institution": "School of Mathematics, Institute for Advanced Study, Princeton NJ, U.S.A.",
            "department": ""
        }
    ],
    "date_submitted": "2025-09-12T10:16:03Z",
    "date_accepted": "2025-09-12T10:16:03Z",
    "date_published": "2025-09-15T07:00:00Z",
    "render_galley": null,
    "galleys": [
        {
            "label": "",
            "type": "pdf",
            "path": "https://journalpub.escholarship.org/combinatorial_theory/article/65000/galley/49810/download/"
        }
    ]
}