{
  "abstract": "Let $u=(u_n)_{n\\ge0}$ be a binary sequence, and define \\[ X_0=1,\\qquad X_{n+1}=2X_n+u_n \\qquad(n\\ge0). \\] Let $A_u$ be the set of all nonempty finite subset sums of the sequence $(X_n)$, and let $L_u(N)$ denote the maximum length of an arithmetic progression contained in $A_u\\cap[1,N]$. We prove that there are absolute constants $c>0$ and $N_0\\geq 1$ such that, for every binary sequence $u$, \\[ L_u(N)\\ge \\exp\\!\\left(c\\sqrt{\\frac{\\log N}{\\log\\log N}}\\right) \\] for all $N\\geq N_0$. Moreover, if the random variables $u_n$ are independent and uniformly distributed on $\\{0,1\\}$, then, almost surely, \\[ L_u(N)\\ll_u N^{2/3}\\exp\\!\\left(C\\sqrt{\\log N\\log\\log N}\\right), \\] where $C>0$ is an absolute constant. Furthermore, every eventually periodic binary sequence satisfies $L_u(N)\\gg_u N^{1/2}$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.33067",
  "author_total": 3,
  "authors": [
    {
      "name": "Norbert Hegyvári"
    },
    {
      "name": "Thang Pham"
    },
    {
      "name": "Boqing Xue"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/9js6n4kvjd3xj.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.33067",
  "fulltext_url": null,
  "id": "9js6n4kvjd3xj",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.33067",
  "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"
    ],
    [
      "openalex",
      "2602|Analytic Number Theory Research"
    ]
  ],
  "openalex": "W7214766089",
  "pdf_url": "https://arxiv.org/pdf/2609.33067",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-27",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/9js6n4kvjd3xj.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:17Z",
      "native_id": "2609.33067",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.33067"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Arithmetic Progressions in a Random Binary Subset-Sum Set",
  "title_text": "Arithmetic Progressions in a Random Binary Subset-Sum Set",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:17Z",
  "url": "https://research.draconitic.com/paper/9js6n4kvjd3xj",
  "venue": "arXiv",
  "versions": []
}