{"paper":{"title":"Affine Approximation in Finite Nagata Dimension and Applications to Lipschitz-free spaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.MG"],"primary_cat":"math.FA","authors_text":"Andr\\'es Quilis, Anton\\'in Prochazka, Colin Petitjean, Mingu Jung","submitted_at":"2026-06-10T06:55:49Z","abstract_excerpt":"We show that if $M$ is a metric space of Nagata dimension at most $d$, then there exists an atlas on $M$ modeled on $\\mathbb R^d$ such that every Lipschitz map $f:M\\to Y$ (with values in an arbitrary Banach space $Y$) can be uniformly approximated by maps that are affine, and thus $\\mathcal{C}^1$-smooth, with respect to this atlas. The construction relies on random metric partitions and stochastic retractions inside Lipschitz-free spaces. As an application, we introduce approximate continuous upper gradient $X$-structures (ACUG $X$-structures) on metric spaces and prove that every space of fin"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.11723","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/2606.11723/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"}