{
  "abstract": "Let $p$ be an odd prime and let $S\\subseteq\\mathbb{F}_p^2$ have cardinality divisible by $p$. We prove that, for all sufficiently large primes $p$, no such set has exactly four special directions, and obtain a conditional extension to larger numbers of special directions under an affine-independence assumption on the corresponding projection functions. A separate second-moment argument shows more generally that, for every fixed $k\\ge4$, no subset of $\\mathbb{F}_p^2$ has exactly $k$ special directions once $p$ is sufficiently large. In fact, the result holds uniformly for $k$ up to a positive constant times $\\sqrt p/\\log p$. In contrast, for multisets every prescribed collection of at most $p$ directions can occur as the set of special directions of a $\\{0,1,2\\}$-valued multiset on $\\mathbb{F}_p^2$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.21779",
  "author_total": 3,
  "authors": [
    {
      "name": "Luca Ghidelli"
    },
    {
      "name": "Gergely Kiss"
    },
    {
      "name": "Gábor Somlai"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/e7pmxx3athm40.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.21779",
  "fulltext_url": null,
  "id": "e7pmxx3athm40",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.21779",
  "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"
    ],
    [
      "openalex",
      "2607|Limits and Structures in Graph Theory"
    ]
  ],
  "openalex": "W7213886812",
  "pdf_url": "https://arxiv.org/pdf/2609.21779",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-18",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/e7pmxx3athm40.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:07Z",
      "native_id": "2609.21779",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.21779"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Non-existence of sets with few special directions",
  "title_text": "Non-existence of sets with few special directions",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:07Z",
  "url": "https://research.draconitic.com/paper/e7pmxx3athm40",
  "venue": "arXiv",
  "versions": []
}