{
  "abstract": "In this paper, we derive thresholds and fluctuations for arithmetic progressions with prescribed color patterns in sparse random colorings of $[n]:=\\{1, 2, \\ldots, n\\}$, where each element of $[n]$ is colored independently according to a given probability vector. For any admissible ordered palette of colors, we determine the full multi-parameter threshold region for the appearance of a colorful arithmetic progression. The threshold is governed by two competing mechanisms: a global first-moment condition and a local color-availability condition, resulting in a polyhedral satisfiability region, with a piecewise-polyhedral threshold surface. In the satisfiability region we establish asymptotic normality for the number of colorful arithmetic progressions of a given length, with an explicit rate of convergence in Wasserstein distance. On the threshold surface, we identify three distinct asymptotic regimes: Poisson, compound Poisson with mixed Poisson jumps, and compound Poisson with uniform jumps, after an appropriate normalization. These results provide a complete description of the threshold and fluctuation behavior of general colored arithmetic progressions under sparse random colorings, in a unified framework that interpolates between classical uncolored/monochromatic progressions in binomial random subsets and multicolored, including rainbow, arithmetic progressions.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.15086",
  "author_total": 3,
  "authors": [
    {
      "name": "Bhaswar B. Bhattacharya"
    },
    {
      "name": "Sanchayan Bhowal"
    },
    {
      "name": "Atmadeep Sengupta"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/9dxtv0q9ry3sf.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.15086",
  "fulltext_url": null,
  "id": "9dxtv0q9ry3sf",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.15086",
  "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.CO"
    ],
    [
      "arxiv",
      "math.NT"
    ],
    [
      "arxiv",
      "math.PR"
    ],
    [
      "openalex",
      "2607|Limits and Structures in Graph Theory"
    ]
  ],
  "openalex": "W7213330687",
  "pdf_url": "https://arxiv.org/pdf/2609.15086",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-28",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/9dxtv0q9ry3sf.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:17Z",
      "native_id": "2609.15086",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.15086"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Thresholds and Fluctuations for Colorful Arithmetic Progressions in Sparse Random Colorings",
  "title_text": "Thresholds and Fluctuations for Colorful Arithmetic Progressions in Sparse Random Colorings",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:17Z",
  "url": "https://research.draconitic.com/paper/9dxtv0q9ry3sf",
  "venue": "arXiv",
  "versions": []
}