{"paper":{"title":"Balanced intersection size distributions in projective planes","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.NT"],"primary_cat":"math.CO","authors_text":"Zolt\\'an L\\'or\\'ant Nagy, Zsuzsa Weiner","submitted_at":"2026-05-22T13:55:46Z","abstract_excerpt":"Given a point set $S$ in a projective plane $\\Pi_q$ of order $q$, each line $\\ell$ determines a secant size\n  $|S\\cap \\ell|$. We study how balanced the secant-size distribution can be for the line set $\\mathcal{L}$ of the plane, in other words, how many lines must share the same secant size.\n  We show that $\\min_{ S\\subseteq \\Pi_q} \\max_k |\\{\\ell\\in \\mathcal{L}: |\\ell\\cap S|=k\\}|=\\Theta(q^{3/2}).$\n  This shows a large contrast with the case of real projective (or affine) plane, where $\\max_{k>1} |\\{\\ell\\in~ \\mathcal{L}: |\\ell\\cap S|=k\\}|$ is always at least the third of $|\\{\\ell\\in \\mathcal{L}"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2605.23644","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2605.23644/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}