{
  "abstract": "We prove that every set $\\mathcal{B}\\subseteq\\mathbb{Z}_{\\ge0}$ for which $\\mathbb{P}+\\mathcal{B}$ contains all sufficiently large integers satisfies $B(X)\\gg\\log X\\log\\log X$, where $B(X)=\\#(\\mathcal{B}\\cap[0,X])$ and the implied constant is absolute. This proves a quantitative form of Ruzsa's Conjecture 2 and gives a negative answer to the $O(\\log X)$ question in Erdős Problem 32. The bound follows from a more general result: for an arbitrary set $\\mathcal{B}\\subseteq\\mathbb{Z}_{\\ge0}$ and sufficiently large $X$, we construct a block of $\\lfloor c\\log X\\log\\log X/(B(X)+1)\\rfloor$ consecutive integers in $[X/2,X]$ outside $\\mathbb{P}+\\mathcal{B}$, provided this length is positive. As an application, we obtain $G_{\\mathcal R}(X)\\gg\\log\\log X$ for the longest block in $[1,X]$ containing no Romanoff number $p+2^n$, improving the bound $G_{\\mathcal R}(X)\\gg\\log\\log X/\\log\\log\\log X$ of Kalmynin and Konyagin. The argument also applies to $p+a^n$ for every fixed integer $a\\ge2$. The key step is a lemma on large shifts, proved using truncated products of divisor sums and the removal of auxiliary primes dividing differences of shifts.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.14127",
  "author_total": 1,
  "authors": [
    {
      "name": "Artyom Radomskii"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/5fkw5g2g5pxxj.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.14127",
  "fulltext_url": null,
  "id": "5fkw5g2g5pxxj",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.14127",
  "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.NT"
    ],
    [
      "openalex",
      "2602|Analytic Number Theory Research"
    ]
  ],
  "openalex": "W7221823695",
  "pdf_url": "https://arxiv.org/pdf/2610.14127",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-19",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/5fkw5g2g5pxxj.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:30:57Z",
      "native_id": "2610.14127",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.14127"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "A lower bound for additive complements to the primes",
  "title_text": "A lower bound for additive complements to the primes",
  "type": "preprint",
  "updated_at": "2026-11-05T15:20:22Z",
  "url": "https://research.draconitic.com/paper/5fkw5g2g5pxxj",
  "venue": "arXiv",
  "versions": []
}