{
  "abstract": "Let $\\{a_k\\}_{k\\ge1}$ be an infinite strictly increasing sequence of positive integers with $a_k\\mid a_{k+1}$ for every integer $k\\ge1$. Dai and Pan proved that the set of integers of the form $p^2+q^2+a_k+a_l$, where $p,q$ are primes and $k,l\\ge1$, has positive lower asymptotic density if $\\limsup_{k\\to\\infty}\\frac{\\log a_k}{k}<\\infty$ and $\\sup_{k\\ge1}\\sum_{d\\mid a_k}\\frac1d<\\infty$. They also asked whether the latter condition is necessary. In this paper, we show that this condition is necessary by proving that if $\\sup_{k\\ge1}\\sum_{d\\mid a_k}\\frac1d=\\infty$, then this set has asymptotic density zero.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.28237",
  "author_total": 2,
  "authors": [
    {
      "name": "Xiao-Hui Yan"
    },
    {
      "name": "Ya-Ting Xu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/960jnthn9a4pq.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.28237",
  "fulltext_url": null,
  "id": "960jnthn9a4pq",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.28237",
  "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": "W7223125013",
  "pdf_url": "https://arxiv.org/pdf/2610.28237",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-20",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/960jnthn9a4pq.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:34:08Z",
      "native_id": "2610.28237",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.28237"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "A necessary condition for a Romanoff type theorem",
  "title_text": "A necessary condition for a Romanoff type theorem",
  "type": "preprint",
  "updated_at": "2026-10-30T12:34:08Z",
  "url": "https://research.draconitic.com/paper/960jnthn9a4pq",
  "venue": "arXiv",
  "versions": []
}