{"paper":{"title":"Densest packings of translates of strings and layers of balls","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.MG","authors_text":"A. Heppes, E. Makai Jr, K. B\\\"or\\\"oczky","submitted_at":"2017-06-16T14:20:43Z","abstract_excerpt":"Let $L \\subset {\\Bbb R}^3$ be the union of unit balls, whose centres lie on the $z$-axis, and are equidistant with distance $2d \\in [2, 2\\sqrt{2}]$. Then a packing of unit balls in ${\\Bbb R}^3$ consisting of translates of $L$ has a density at most $\\pi /(3d\\sqrt{3-d^2})$, with equality for a certain lattice packing of unit balls. Let $L \\subset {\\Bbb R}^4$ be the union of unit balls, whose centres lie on the $x_3x_4$ coordinate plane, and form either a square lattice or a regular triangular lattice, of edge length $2$. Then a packing of unit balls in ${\\Bbb R}^4$ consisting of translates of $L"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1706.05282","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":""},"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"}