{
  "abstract": "Let $\\mathbb N$ denote the set of nonnegative integers. For an integer $k>1$ and a set $A\\subseteq\\mathbb N$, let $R_{1,k}(A,n)$ denote the number of solutions of $n=a_1+ka_2$ with $a_1,a_2\\in A$. For integers $k>1$ and $t\\ge1$, Yang and Chen defined $f_k(t)$ to be the number of sets $A\\subseteq\\mathbb N$ for which \\[ R_{1,k}(A,n)=R_{1,k}(\\mathbb N\\setminus A,n) \\] for all integers $n\\ge t$, and asked whether $f_k(t)$ and $f_l(t)$ are eventually equal for any integers $k,l>1$. We prove that \\[ f_k(t)\\asymp_k \\frac{2^t}{t^{k/2}}, \\] which gives a negative answer to their problem.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.20385",
  "author_total": 3,
  "authors": [
    {
      "name": "Shuang-Shuang Li"
    },
    {
      "name": "Ya-Ting Xu"
    },
    {
      "name": "Xiao-Hui Yan"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/14fftr7dm1yy9.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.20385",
  "fulltext_url": null,
  "id": "14fftr7dm1yy9",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.20385",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.NT"
    ],
    [
      "openalex",
      "2602|Analytic Number Theory Research"
    ]
  ],
  "openalex": "W7213579411",
  "pdf_url": "https://arxiv.org/pdf/2609.20385",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-17",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/14fftr7dm1yy9.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:19Z",
      "native_id": "2609.20385",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.20385"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "A problem of Yang and Chen on weighted representation functions",
  "title_text": "A problem of Yang and Chen on weighted representation functions",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:19Z",
  "url": "https://research.draconitic.com/paper/14fftr7dm1yy9",
  "venue": "arXiv",
  "versions": []
}