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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"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1907.06903","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2019-07-16T09:12:06Z","cross_cats_sorted":["math.CA"],"title_canon_sha256":"d3f7c4662c7fd4875ab11a2e43e9d5fac786022f336d756c870820538e231c5d","abstract_canon_sha256":"6045f0d62b054857dc4e4216a128e9eb8fbd7204c6efb1dd0d671fbbfb9e4fd4"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:40:28.740583Z","signature_b64":"SA+r+nkXpXfTgkEx2jnYGjtbH1bcUR+g6VtJySO/M5O0HvK7i9ZFj1dKpJiFcZ0dbhKcCWifrhjn/nWgmqX+CQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"7921144b0461fe3d074739a88451cb84601f995b1a6204b36cb4b04616357838","last_reissued_at":"2026-05-17T23:40:28.739907Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:40:28.739907Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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. 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