{"paper":{"title":"A simple proof of a reverse Minkowski theorem for integral lattices","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.NT"],"primary_cat":"math.MG","authors_text":"Noah Stephens-Davidowitz, Oded Regev","submitted_at":"2023-06-06T14:11:42Z","abstract_excerpt":"We prove that for any integral lattice $\\mathcal{L} \\subset \\mathbb{R}^n$ (that is, a lattice $\\mathcal{L}$ such that the inner product $\\langle \\mathbf{y}_1,\\mathbf{y}_2 \\rangle$ is an integer for all $\\mathbf{y}_1, \\mathbf{y}_2 \\in \\mathcal{L}$) and any positive integer $k$,\n  \\[\n  |\\{ \\mathbf{y} \\in \\mathcal{L} \\ : \\ \\|\\mathbf{y}\\|^2 = k\\}| \\leq 2 \\binom{n+2k-2}{2k-1}\n  \\; ,\n  \\]\n  giving a nearly tight reverse Minkowski theorem for integral lattices."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2306.03697","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/2306.03697/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"}