{
  "abstract": "We investigate a recently introduced generalisation of determined directions in affine planes. A direction $(d)$ in an affine plane is a point of the line at infinity in the projective completion, and hence corresponds to a parallel class. A (multi)set $S$ of points is equidistributed from direction $(d)$ if all lines from that parallel class intersect $S$ in the same number of points. In this paper, we give a construction of point (multi)sets that are inequidistributed from exactly 3 directions in any finite translation plane, and prove that all such (multi)sets arise from this construction, generalising a result of Kiss and Somlai in Desarguesian planes of prime order. We also use ideas from algebraic graph theory to prove results on directions associated with a pair of point sets $S$ and $T$. If for every direction $(d)$ either $S$ or $T$ is equidistributed from $(d)$, we prove that $|S \\cap T| = |S||T|/q^2$, where $q$ is the order of the affine plane. In the appendix, we give a combinatorial alternative approach to proving this equality, which can be of independent interest. We also introduce the notion of directions cross-determined by a pair of sets $S$ and $T$, and prove that $|S| |T| \\leq q^2$ if $(S,T)$ cross-determines at most half of the directions, where equality forces $S = T$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.40023",
  "author_total": 3,
  "authors": [
    {
      "name": "Sam Adriaensen"
    },
    {
      "name": "Bence Csajbók"
    },
    {
      "name": "Zsuzsa Weiner"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/a9s9dmkw49x17.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.40023",
  "fulltext_url": null,
  "id": "a9s9dmkw49x17",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.40023",
  "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"
    ],
    [
      "openalex",
      "2607|Advanced Combinatorial Mathematics"
    ]
  ],
  "openalex": "W7215106320",
  "pdf_url": "https://arxiv.org/pdf/2609.40023",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-30",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/a9s9dmkw49x17.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:49Z",
      "native_id": "2609.40023",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.40023"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "On equidistributed directions in finite affine planes",
  "title_text": "On equidistributed directions in finite affine planes",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:49Z",
  "url": "https://research.draconitic.com/paper/a9s9dmkw49x17",
  "venue": "arXiv",
  "versions": []
}