{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2011:7JQBNOXDAJNBDWDNT5H2ACESDM","short_pith_number":"pith:7JQBNOXD","canonical_record":{"source":{"id":"1107.0424","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2011-07-03T02:46:47Z","cross_cats_sorted":["math.DS"],"title_canon_sha256":"54b2abc65bfc2f2c30d84358a4d99e0c9b235d0be27e77a3acfa373c879d9e32","abstract_canon_sha256":"7a462d740dd4d9f449984dadc9f190f13b242842f97c7cf1198a1cdfac5751b1"},"schema_version":"1.0"},"canonical_sha256":"fa6016bae3025a11d86d9f4fa008921b0106f0d2b1f77ffabac02afb62e36cf2","source":{"kind":"arxiv","id":"1107.0424","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1107.0424","created_at":"2026-05-18T04:18:53Z"},{"alias_kind":"arxiv_version","alias_value":"1107.0424v1","created_at":"2026-05-18T04:18:53Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1107.0424","created_at":"2026-05-18T04:18:53Z"},{"alias_kind":"pith_short_12","alias_value":"7JQBNOXDAJNB","created_at":"2026-05-18T12:26:22Z"},{"alias_kind":"pith_short_16","alias_value":"7JQBNOXDAJNBDWDN","created_at":"2026-05-18T12:26:22Z"},{"alias_kind":"pith_short_8","alias_value":"7JQBNOXD","created_at":"2026-05-18T12:26:22Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2011:7JQBNOXDAJNBDWDNT5H2ACESDM","target":"record","payload":{"canonical_record":{"source":{"id":"1107.0424","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2011-07-03T02:46:47Z","cross_cats_sorted":["math.DS"],"title_canon_sha256":"54b2abc65bfc2f2c30d84358a4d99e0c9b235d0be27e77a3acfa373c879d9e32","abstract_canon_sha256":"7a462d740dd4d9f449984dadc9f190f13b242842f97c7cf1198a1cdfac5751b1"},"schema_version":"1.0"},"canonical_sha256":"fa6016bae3025a11d86d9f4fa008921b0106f0d2b1f77ffabac02afb62e36cf2","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:18:53.999117Z","signature_b64":"eJdTotHMguo1Xm8KppPRz96Kglam/BB29X6RaV+X1cjZqh592RyFxevl5s+x3NW6MSHJK8KQREPCvOkWk7t2Cw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"fa6016bae3025a11d86d9f4fa008921b0106f0d2b1f77ffabac02afb62e36cf2","last_reissued_at":"2026-05-18T04:18:53.998651Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:18:53.998651Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1107.0424","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T04:18:53Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"5wtPWRQ5LxZaON3rjDSF31N0kS6YpNELZ7sIZJGUI8zPIY7+kFIMngnI8A0WvNo9kaTq3OJiRjo7UL451gMjAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-24T12:10:24.328538Z"},"content_sha256":"a101531e74d67f822fc69e41c4febd4397feaef65196281cf6d2f16e76f14ba2","schema_version":"1.0","event_id":"sha256:a101531e74d67f822fc69e41c4febd4397feaef65196281cf6d2f16e76f14ba2"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2011:7JQBNOXDAJNBDWDNT5H2ACESDM","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"A generalization of Marstrand's theorem for projections of cartesian products","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.DS"],"primary_cat":"math.CA","authors_text":"Carlos Gustavo Moreira, Jorge Erick L\\'opez","submitted_at":"2011-07-03T02:46:47Z","abstract_excerpt":"We prove the following variant of Marstrand's theorem about projections of cartesian products of sets:\n  Let $K_1,...,K_n$ Borel subsets of $\\mathbb R^{m_1},... ,\\mathbb R^{m_n}$ respectively, and $\\pi:\\mathbb R^{m_1}\\times...\\times\\mathbb R^{m_n}\\to\\mathbb R^k$ be a surjective linear map. We set $$\\mathfrak{m}:=\\min\\{\\sum_{i\\in I}\\dim_H(K_i) + \\dim\\pi(\\bigoplus_{i\\in I^c}\\mathbb R^{m_i}), I\\subset\\{1,...,n\\}, I\\ne\\emptyset\\}.$$ Consider the space $\\Lambda_m=\\{(t,O), t\\in\\mathbb R, O\\in SO(m)\\}$ with the natural measure and set $\\Lambda=\\Lambda_{m_1}\\times...\\times\\Lambda_{m_n}$. For every $\\l"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1107.0424","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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T04:18:53Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"IeurbLf9Q6zbwIgMDUbEF17mNlRFG8pg9zsh33aEDsqgWy1dQ3cPrIvKghTofF+w/VHazr6qvhK08qetD6RmBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-24T12:10:24.329168Z"},"content_sha256":"4b45d69c098a07c8a6a5ad2c75672ed81c74623508e6b69ace2b484078bfb85e","schema_version":"1.0","event_id":"sha256:4b45d69c098a07c8a6a5ad2c75672ed81c74623508e6b69ace2b484078bfb85e"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/7JQBNOXDAJNBDWDNT5H2ACESDM/bundle.json","state_url":"https://pith.science/pith/7JQBNOXDAJNBDWDNT5H2ACESDM/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/7JQBNOXDAJNBDWDNT5H2ACESDM/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-05-24T12:10:24Z","links":{"resolver":"https://pith.science/pith/7JQBNOXDAJNBDWDNT5H2ACESDM","bundle":"https://pith.science/pith/7JQBNOXDAJNBDWDNT5H2ACESDM/bundle.json","state":"https://pith.science/pith/7JQBNOXDAJNBDWDNT5H2ACESDM/state.json","well_known_bundle":"https://pith.science/.well-known/pith/7JQBNOXDAJNBDWDNT5H2ACESDM/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:7JQBNOXDAJNBDWDNT5H2ACESDM","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"7a462d740dd4d9f449984dadc9f190f13b242842f97c7cf1198a1cdfac5751b1","cross_cats_sorted":["math.DS"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2011-07-03T02:46:47Z","title_canon_sha256":"54b2abc65bfc2f2c30d84358a4d99e0c9b235d0be27e77a3acfa373c879d9e32"},"schema_version":"1.0","source":{"id":"1107.0424","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1107.0424","created_at":"2026-05-18T04:18:53Z"},{"alias_kind":"arxiv_version","alias_value":"1107.0424v1","created_at":"2026-05-18T04:18:53Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1107.0424","created_at":"2026-05-18T04:18:53Z"},{"alias_kind":"pith_short_12","alias_value":"7JQBNOXDAJNB","created_at":"2026-05-18T12:26:22Z"},{"alias_kind":"pith_short_16","alias_value":"7JQBNOXDAJNBDWDN","created_at":"2026-05-18T12:26:22Z"},{"alias_kind":"pith_short_8","alias_value":"7JQBNOXD","created_at":"2026-05-18T12:26:22Z"}],"graph_snapshots":[{"event_id":"sha256:4b45d69c098a07c8a6a5ad2c75672ed81c74623508e6b69ace2b484078bfb85e","target":"graph","created_at":"2026-05-18T04:18:53Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"We prove the following variant of Marstrand's theorem about projections of cartesian products of sets:\n  Let $K_1,...,K_n$ Borel subsets of $\\mathbb R^{m_1},... ,\\mathbb R^{m_n}$ respectively, and $\\pi:\\mathbb R^{m_1}\\times...\\times\\mathbb R^{m_n}\\to\\mathbb R^k$ be a surjective linear map. We set $$\\mathfrak{m}:=\\min\\{\\sum_{i\\in I}\\dim_H(K_i) + \\dim\\pi(\\bigoplus_{i\\in I^c}\\mathbb R^{m_i}), I\\subset\\{1,...,n\\}, I\\ne\\emptyset\\}.$$ Consider the space $\\Lambda_m=\\{(t,O), t\\in\\mathbb R, O\\in SO(m)\\}$ with the natural measure and set $\\Lambda=\\Lambda_{m_1}\\times...\\times\\Lambda_{m_n}$. For every $\\l","authors_text":"Carlos Gustavo Moreira, Jorge Erick L\\'opez","cross_cats":["math.DS"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2011-07-03T02:46:47Z","title":"A generalization of Marstrand's theorem for projections of cartesian products"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1107.0424","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:a101531e74d67f822fc69e41c4febd4397feaef65196281cf6d2f16e76f14ba2","target":"record","created_at":"2026-05-18T04:18:53Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"7a462d740dd4d9f449984dadc9f190f13b242842f97c7cf1198a1cdfac5751b1","cross_cats_sorted":["math.DS"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2011-07-03T02:46:47Z","title_canon_sha256":"54b2abc65bfc2f2c30d84358a4d99e0c9b235d0be27e77a3acfa373c879d9e32"},"schema_version":"1.0","source":{"id":"1107.0424","kind":"arxiv","version":1}},"canonical_sha256":"fa6016bae3025a11d86d9f4fa008921b0106f0d2b1f77ffabac02afb62e36cf2","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"fa6016bae3025a11d86d9f4fa008921b0106f0d2b1f77ffabac02afb62e36cf2","first_computed_at":"2026-05-18T04:18:53.998651Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:18:53.998651Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"eJdTotHMguo1Xm8KppPRz96Kglam/BB29X6RaV+X1cjZqh592RyFxevl5s+x3NW6MSHJK8KQREPCvOkWk7t2Cw==","signature_status":"signed_v1","signed_at":"2026-05-18T04:18:53.999117Z","signed_message":"canonical_sha256_bytes"},"source_id":"1107.0424","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:a101531e74d67f822fc69e41c4febd4397feaef65196281cf6d2f16e76f14ba2","sha256:4b45d69c098a07c8a6a5ad2c75672ed81c74623508e6b69ace2b484078bfb85e"],"state_sha256":"41c0cea182f099aace47d1ec74ec4b4520795cbe337d46e10419c2448d8ee545"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"rwuZPfNPzOzQy1lh+Bu9bMeYlrW3mJQA2kFhfQXFPzzsBOg5ZEBR2rxrl//Z+kJT8hXqzhvLBU+miopPsi+BCA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-24T12:10:24.332842Z","bundle_sha256":"0badd44b0e8870c22cdd2c889972ac6b9a6f22272cca06ae86fec163767c73f7"}}