{"paper":{"title":"Horizontal Kakeya maximal operators in finite Heisenberg groups: Exact exponents and applications","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.CA","math.GR","math.NT"],"primary_cat":"math.CO","authors_text":"Andrea Pinamonti, Boqing Xue, Dung The Tran, Thang Pham","submitted_at":"2026-03-02T17:28:02Z","abstract_excerpt":"Let $q$ be an odd prime power. We study Kakeya maximal operators associated with horizontal lines in the finite Heisenberg groups $\\mathbb H_n(\\mathbb F_q)$. Our principal object is the refined-direction maximal operator, whose parameter records the projective horizontal direction together with the central homogeneous coordinate determined by horizontality. In rank one, we prove \\[ \\|M_{\\mathbb H_1}^{\\mathrm{rd}}F\\|_{\\ell^2(\\mathcal D_1)} \\lesssim q^{\\frac{1}{2}} \\|F\\|_{\\ell^2(\\mathbb H_1(\\mathbb F_q))}, \\] where the exponent $\\frac{1}{2}$ is sharp. Combining this estimate with endpoint bounds"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2603.02111","kind":"arxiv","version":2},"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/2603.02111/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"}