{
  "abstract": "Let $A$ be an $n$-element set of positive real numbers, let $FS(A)$ be its set of subset sums, and put $T_n=\\binom{n+1}{2}$. For every fixed integer $C\\geq-1$ and all sufficiently large $n$, we classify the sets satisfying $$ |FS(A)|\\leq T_n+n+C+1. $$ Each such set is commensurable. Its unique primitive integer normalisation $B$ either satisfies $\\sum B\\leq T_n+n+C$ or belongs to an explicit exceptional family specified by a missing element $m\\in\\{1,2\\}$ and an integer partition of $C+m$ or $C+m+1$. If $P$ denotes the partition function, the exceptional family has exactly $$ P(C+1)+2P(C+2)+P(C+3) $$ primitive dilation classes. We also prove a local inverse theorem for bounded increment excess. If, for sufficiently large $i$, adjoining the largest element to the preceding $i-1$ elements creates only $i+e$ new subset sums, where $e$ is bounded, then the $i$-element set is a dilation of $[1,i+e]_{\\mathbb{Z}}$ with exactly $e$ elements deleted. Conversely, every such deletion pattern has increment excess $e$. The proof combines a stabiliser argument in $\\mathbb{R}/x\\mathbb{Z}$, Kneser's theorem, a quadratic subset-sum bound, and endpoint propagation. These arguments also give effective commensurability and a finite-state encoding. Together with earlier results for $C\\leq-2$, this completes the eventual fixed-defect classification for every integer $C$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.00044",
  "author_total": 1,
  "authors": [
    {
      "name": "Lizhong Chen"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/80tavnk3pa7g3.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.00044",
  "fulltext_url": null,
  "id": "80tavnk3pa7g3",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.00044",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ],
    [
      "arxiv",
      "math.NT"
    ],
    [
      "openalex",
      "2607|Advanced Combinatorial Mathematics"
    ]
  ],
  "openalex": "W7206164283",
  "pdf_url": "https://arxiv.org/pdf/2609.00044",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-14",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/80tavnk3pa7g3.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:29Z",
      "native_id": "2609.00044",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.00044"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Fixed-Defect Inverse Theorems for Subset Sums",
  "title_text": "Fixed-Defect Inverse Theorems for Subset Sums",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:29Z",
  "url": "https://research.draconitic.com/paper/80tavnk3pa7g3",
  "venue": "arXiv",
  "versions": []
}