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The condition $\\textb"},"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":"2411.17018","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2024-11-26T01:06:10Z","cross_cats_sorted":[],"title_canon_sha256":"9818ae2e82d6ec510c58deed82b934ba4119d11399fb848f8e7e1dbefa6069ed","abstract_canon_sha256":"7e01a48024b07e07ce74530b5e58b7ee6c1454bcb0e99ccbda457c0698b8092d"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T09:40:26.151634Z","signature_b64":"YUxC/RDCOCr1VLVzBY8vaNFvgCL7N59Y+KN9xmhmWetdDWZE+RJ9JxNIR/FAItt5ni7xPYS/SD3QMxvLfr8vAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"53b6d2806d45603d87abf36b07c3d825b3bf4afef6fd0cdae8ce1d6ebad4f9d4","last_reissued_at":"2026-07-05T09:40:26.151102Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T09:40:26.151102Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The Hausdorff measure and uniform fibre conditions for Bara\\'nski carpet","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Hua Qiu, Qi Wang","submitted_at":"2024-11-26T01:06:10Z","abstract_excerpt":"For a self-affine carpet $K$ of Bara\\'{n}ski, we establish a dichotomy:\n  $\n  \\text{either }\\quad 0<\\mathcal{H}^{\\dim_{\\text{H}} K}(K)<+\\infty \\quad\\text{ or } \\quad\\mathcal{H}^{\\dim_{\\text{H}} K}(K)=+\\infty.\n  $\n  We introduce four types of uniform fibre condition for $K$: Hausdorff ($\\textbf{u.f.H}$), Box ($\\textbf{u.f.B}$), Assouad ($\\textbf{u.f.A}$), and Lower ($\\textbf{u.f.L}$), which are progressively stronger, with\n  $\n  \\textbf{u.f.L} \\Longrightarrow \\textbf{u.f.A} \\Longrightarrow \\textbf{u.f.B} \\Longrightarrow \\textbf{u.f.H},\n  $\n  and each implication is strict. 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