{"paper":{"title":"Quantitative Alberti representations in spaces of bounded geometry","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CA"],"primary_cat":"math.MG","authors_text":"Tuomas Orponen","submitted_at":"2019-07-16T09:12:06Z","abstract_excerpt":"A metric measure space $(X,d,\\mu)$ is said to be $A_{\\infty}$ on curves if there exist constants $\\tau < 1$ and $\\theta > 0$ with the following property. For every $x \\in X$, $0 < r \\leq \\mathrm{diam}(X)$, and a Borel set $S \\subset B(x,r)$ with $\\mu(S) > \\tau \\mu(B(x,r))$, there exists a continuum $\\gamma \\subset X$ of length $\\leq r$ satisfying $\\mathcal{H}^{1}_{\\infty}(\\gamma \\cap S) \\geq \\theta r$.\n  I first observe that spaces of $Q$-bounded geometry, $Q > 1$, are $A_{\\infty}$ on curves. Then, I show that any complete, doubling, and quasiconvex space $(X,d,\\mu)$ which is $A_{\\infty}$ on c"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1907.06903","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"}