{"paper":{"title":"Quantitative Estimates on the Singular Sets of Alexandrov Spaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.MG"],"primary_cat":"math.DG","authors_text":"Aaron Naber, Nan Li","submitted_at":"2019-12-08T04:51:47Z","abstract_excerpt":"Let $X\\in\\text{Alex}\\,^n(-1)$ be an $n$-dimensional Alexandrov space with curvature $\\ge -1$. Let the $r$-scale $(k,\\epsilon)$-singular set $\\mathcal S^k_{\\epsilon,\\,r}(X)$ be the collection of $x\\in X$ so that $B_r(x)$ is not $\\epsilon r$-close to a ball in any splitting space $\\mathbb R^{k+1}\\times Z$. We show that there exists $C(n,\\epsilon)>0$ and $\\beta(n,\\epsilon)>0$, independent of the volume, so that for any disjoint collection $\\big\\{B_{r_i}(x_i):x_i\\in \\mathcal S_{\\epsilon,\\,\\beta r_i}^k(X)\\cap B_1, \\,r_i\\le 1\\big\\}$, the packing estimate $\\sum r_i^k\\le C$ holds. Consequently, we obt"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1912.03615","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/1912.03615/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"}