{"paper":{"title":"Asymptotic-Type Dimension Bounds through Combinatorial Approaches","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.MG","authors_text":"Jing Yu, Xingyu Zhu","submitted_at":"2024-11-25T18:45:03Z","abstract_excerpt":"We develop a probabilistic framework for large-scale dimension bounds in metric geometry, based on padded decompositions, randomized ball carving on net graphs, and the Lov\\'asz Local Lemma. For metric measure spaces with volume doubling constant $C_{\\mathsf D}$, we prove the sharp bound $\\mathrm{asdim}_{AN}(X)\\le \\mathrm{dim}_{AN}(X)\\le \\lfloor{\\log_2 C_{\\mathsf D}}\\rfloor$. In particular, if $(M,g)$ is a complete Riemannian $n$-manifold with $\\mathrm{Ric}_g\\ge 0$, then $\\mathrm{asdim}(M)\\le n$, thereby settling a question of Papasoglu on manifolds with nonnegative Ricci curvature. We also sh"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2411.16660","kind":"arxiv","version":6},"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/2411.16660/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"}