{"paper":{"title":"Geometry of Rounding: Near Optimal Bounds and a New Neighborhood Sperner's Lemma","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["cs.CG","cs.DM"],"primary_cat":"cs.CC","authors_text":"A. Pavan, Jamie Radcliffe, Jason Vander Woude, N. V. Vinodchandran, Peter Dixon","submitted_at":"2023-04-10T19:51:14Z","abstract_excerpt":"A partition $\\mathcal{P}$ of $\\mathbb{R}^d$ is called a $(k,\\varepsilon)$-secluded partition if, for every $\\vec{p} \\in \\mathbb{R}^d$, the ball $\\overline{B}_{\\infty}(\\varepsilon, \\vec{p})$ intersects at most $k$ members of $\\mathcal{P}$. A goal in designing such secluded partitions is to minimize $k$ while making $\\varepsilon$ as large as possible. This partition problem has connections to a diverse range of topics, including deterministic rounding schemes, pseudodeterminism, replicability, as well as Sperner/KKM-type results.\n  In this work, we establish near-optimal relationships between $k"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2304.04837","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/2304.04837/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"}