{
  "abstract": "A subset $A\\subseteq \\mathbb N_0$ is $k$-AP covering if there exists a constant $n_0$ such that for every integer $x>n_0$, there exists $d\\in\\mathbb N_0$ such that $x-d, x-2d,\\dots,x-(k-1)d$ are all in $A$. Disproving a conjecture of Kiss, Sándor, and Yang, we prove that for every integer $k\\geq 6$, there exists a constant $\\varepsilon=\\varepsilon_k>0$ and a $k$-AP covering set $A$ such that $|A\\cap \\{0,1,\\dots,n\\}| < n^{\\frac{k-2}{k-1}-\\varepsilon}$ for all sufficiently large $n$. We also relate this problem to the Arithmetic Kakeya Conjecture by Katz and Tao.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.02041",
  "author_total": 1,
  "authors": [
    {
      "name": "Pitchayut Saengrungkongka"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/7yp1n3qy571dg.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.02041",
  "fulltext_url": null,
  "id": "7yp1n3qy571dg",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.02041",
  "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"
    ],
    [
      "arxiv",
      "math.NT"
    ],
    [
      "openalex",
      "2602|Analytic Number Theory Research"
    ]
  ],
  "openalex": "W7207713886",
  "pdf_url": "https://arxiv.org/pdf/2609.02041",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-02",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/7yp1n3qy571dg.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:18:20Z",
      "native_id": "2609.02041",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.02041"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Sparse $k$-AP Covering Sets and the Arithmetic Kakeya Conjecture",
  "title_text": "Sparse k-AP Covering Sets and the Arithmetic Kakeya Conjecture",
  "type": "preprint",
  "updated_at": "2026-10-30T12:18:20Z",
  "url": "https://research.draconitic.com/paper/7yp1n3qy571dg",
  "venue": "arXiv",
  "versions": []
}