{
  "abstract": "A set $S$ of positive integers is $2$-large if every two-coloring contains arbitrarily long monochromatic arithmetic progressions with common difference in $S$. We prove that every such set meets every nil-Bohr neighborhood of $0$. This proves a conjecture of the first author and gives a sharp answer to a question of Host, Kra, and Maass. We also show that, for every fixed nil-Bohr neighborhood $B$ of $0$ and every $\\varepsilon>0$, restricting the common difference to $B$ increases the extremal number for $k$-term arithmetic progressions by at most a factor of $1+\\varepsilon$, provided $k$ is sufficiently large.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.16605",
  "author_total": 2,
  "authors": [
    {
      "name": "Ryan Alweiss"
    },
    {
      "name": "Sohail Farhangi"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/7ajpsyq4tskf5.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.16605",
  "fulltext_url": null,
  "id": "7ajpsyq4tskf5",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.16605",
  "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.CO"
    ],
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      "arxiv",
      "math.DS"
    ],
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      "arxiv",
      "math.NT"
    ],
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      "2607|Limits and Structures in Graph Theory"
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  "openalex": "W7221805888",
  "pdf_url": "https://arxiv.org/pdf/2610.16605",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-21",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/7ajpsyq4tskf5.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:34:22Z",
      "native_id": "2610.16605",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.16605"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.DS",
      "subfield": "dynamical-systems"
    }
  ],
  "title": "$2$-large sets are sets of nil-Bohr recurrence",
  "title_text": "$2$-large sets are sets of nil-Bohr recurrence",
  "type": "preprint",
  "updated_at": "2026-10-30T12:34:22Z",
  "url": "https://research.draconitic.com/paper/7ajpsyq4tskf5",
  "venue": "arXiv",
  "versions": []
}