{
  "abstract": "Let $h \\geq 2$, and let $\\mathbf{b} = (b_1,\\dots,b_h)\\in \\mathbb{Z}^h$ be a zero-sum vector with nonzero coordinates. For a set $A=\\{a_1<a_2<\\cdots\\}\\subseteq\\mathbb{N}$, let $r_{A,\\mathbf{b}}(n)$ denote the number of $h$-tuples $(x_1,\\ldots,x_h)$ of pairwise distinct elements of $A$ satisfying $b_1x_1+\\cdots+b_hx_h=n$. We study density restrictions on sets $A$ for which these representation counts remain small, obtaining analogues of the classical density theorem for infinite Sidon sets. In the case $\\mathbf{b} = (c_1,-c_1,\\dots,c_k,-c_k)$, we prove that if $A(x)/(x/\\log x)^{1/2k}\\to\\infty$, then $\\frac{1}{x}\\sum_{|n|\\leq x} r_{A,\\mathbf{b}}(n)\\to\\infty$, whereas if $A(x)\\gg x^{1/2k}$, then $\\frac{1}{x}\\sum_{|n|\\leq x} r_{A,\\mathbf{b}}(n)\\gg\\log x$. This recovers Chen's theorem on $B_{2k}$-sequences. For general zero-sum vectors $\\mathbf{b}$, we prove analogous bounds under gap conditions: if $a_{n+1}-a_n=o(n^{h-1}\\log n)$, then $\\frac{1}{x}\\sum_{|n|\\leq x} r_{A,\\mathbf{b}}(n)\\to\\infty$, whereas if $a_{n+1}-a_n\\ll n^{h-1}$, then $\\frac{1}{x}\\sum_{|n|\\leq x} r_{A,\\mathbf{b}}(n)\\gg\\log x$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": "Journal reference: Monatsh. Math. 211 (2026), no. 2, 315-327",
  "arxiv": "2607.20753",
  "author_total": 1,
  "authors": [
    {
      "name": "Christian Táfula"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/db7htd1fv98b9.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2607.20753",
  "fulltext_url": null,
  "id": "db7htd1fv98b9",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2607.20753",
  "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": null,
  "pdf_url": "https://arxiv.org/pdf/2607.20753",
  "pmcid": null,
  "pmid": null,
  "published": "2026-07-22",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/db7htd1fv98b9.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:27Z",
      "native_id": "2607.20753",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2607.20753"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Infinite Sidon-type sets for zero-sum linear forms",
  "title_text": "Infinite Sidon-type sets for zero-sum linear forms",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:27Z",
  "url": "https://research.draconitic.com/paper/db7htd1fv98b9",
  "venue": "arXiv",
  "versions": [
    {
      "doi": "10.1007/s00605-026-02211-4",
      "pid": null,
      "relation": "published-as",
      "title": null,
      "venue": null
    }
  ]
}