{
  "abstract": "A set $A\\subseteq\\mathbb{N}$ is called $\\textit{complete}$ if every sufficiently large integer can be written as a sum of distinct elements of $A$. It is $\\textit{strongly complete}$ if it remains complete after one deletes finitely many elements from it. Building on recent work of Bergelson and Simmons and that of Griesmer, we establish a new strong-completeness criterion exploiting a three-component partition of a given set. As an application, we show that $A$ is strongly complete whenever \\[ \\big|A\\cap(2^k,2^{k+1}]\\big|\\ge5 \\] for every sufficiently large $k\\in\\mathbb{N}$, and \\[ \\sum_{a\\in A}\\|aθ\\|=\\infty, \\quad\\forallθ\\in\\mathbb{R}\\setminus\\mathbb{Z}. \\] In particular, this resolves a 1961 conjecture of Erdős. The new strong-completeness criterion also enables us to make progress on a 1996 problem of Burr, Erdős, Graham, and Li concerning strong completeness of mixed power sets by refining a previous result of Bergelson and Simmons. Besides, we study the polynomially perturbed ray set \\[ \\{\\lfloor tα^n\\rfloor,\\lfloor tα^n\\rfloor+P(n):n\\in\\mathbb{N}\\}, \\] which combines the polynomial set $\\{P(n):n\\in\\mathbb{N}\\}$ and the single-ray set $\\{\\lfloor tα^n\\rfloor:n\\in\\mathbb{N}\\}$ both previously considered by Graham, and show that it is strongly complete for any $t>0$ and $α\\in(0,2)$ and any primitive integer-valued polynomial $P$. The machinery developed for the proof of this result also yields other interesting applications.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2607.14071",
  "author_total": 1,
  "authors": [
    {
      "name": "Steve Fan"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/4qavqms384ngq.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2607.14071",
  "fulltext_url": null,
  "id": "4qavqms384ngq",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2607.14071",
  "language": null,
  "licence": "CC-BY-NC-ND-4.0",
  "licence_label": "CC BY-NC-ND 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-nc-nd/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ],
    [
      "arxiv",
      "math.NT"
    ],
    [
      "openalex",
      "2602|Analytic Number Theory Research"
    ]
  ],
  "openalex": "W7169099643",
  "pdf_url": "https://arxiv.org/pdf/2607.14071",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-13",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/4qavqms384ngq.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:35Z",
      "native_id": "2607.14071",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2607.14071"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Strongly complete sets and a conjecture of Erdős",
  "title_text": "Strongly complete sets and a conjecture of Erdős",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:35Z",
  "url": "https://research.draconitic.com/paper/4qavqms384ngq",
  "venue": "arXiv",
  "versions": []
}