{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2011:WMJ2WD2AU7AXIPPOCOG35J6QPK","short_pith_number":"pith:WMJ2WD2A","canonical_record":{"source":{"id":"1110.6790","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2011-10-31T13:39:21Z","cross_cats_sorted":["math.CO","math.MG"],"title_canon_sha256":"caa947aa04f898d49c206cad785bc1e2c7807fd5f068ab563b40d74d0281101d","abstract_canon_sha256":"f646ba8dd68d8f0edaf5987b13d3db7e509778c49b1176e03bf0f82c261a2e6e"},"schema_version":"1.0"},"canonical_sha256":"b313ab0f40a7c1743dee138dbea7d07aadb9240f1b1f90f25a5367f1aa650f4a","source":{"kind":"arxiv","id":"1110.6790","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1110.6790","created_at":"2026-05-18T04:09:54Z"},{"alias_kind":"arxiv_version","alias_value":"1110.6790v1","created_at":"2026-05-18T04:09:54Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1110.6790","created_at":"2026-05-18T04:09:54Z"},{"alias_kind":"pith_short_12","alias_value":"WMJ2WD2AU7AX","created_at":"2026-05-18T12:26:44Z"},{"alias_kind":"pith_short_16","alias_value":"WMJ2WD2AU7AXIPPO","created_at":"2026-05-18T12:26:44Z"},{"alias_kind":"pith_short_8","alias_value":"WMJ2WD2A","created_at":"2026-05-18T12:26:44Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2011:WMJ2WD2AU7AXIPPOCOG35J6QPK","target":"record","payload":{"canonical_record":{"source":{"id":"1110.6790","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2011-10-31T13:39:21Z","cross_cats_sorted":["math.CO","math.MG"],"title_canon_sha256":"caa947aa04f898d49c206cad785bc1e2c7807fd5f068ab563b40d74d0281101d","abstract_canon_sha256":"f646ba8dd68d8f0edaf5987b13d3db7e509778c49b1176e03bf0f82c261a2e6e"},"schema_version":"1.0"},"canonical_sha256":"b313ab0f40a7c1743dee138dbea7d07aadb9240f1b1f90f25a5367f1aa650f4a","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:09:54.193361Z","signature_b64":"TXCV7kkyRbQxQHgUEXnPFrzIA/YyIe/dyxHccXzdDWLWdbeHXWYgaeUQ9zf9v/hMgSC91sdzUTV/a8f7TGO5Cw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b313ab0f40a7c1743dee138dbea7d07aadb9240f1b1f90f25a5367f1aa650f4a","last_reissued_at":"2026-05-18T04:09:54.192749Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:09:54.192749Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1110.6790","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":"2026-05-18T04:09:54Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"QI5+584BA+bb/6LBqo4mQxEY7Re8KoceLQnDmY1j6RPiK8tqjrc0sf4YH37oX47SYgck9xe7vYcMwIxXqOMzBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-03T18:36:56.009385Z"},"content_sha256":"abebdad18eedceab79c7322d2872572e6e91bf6565e9ac6bb8e0e1caecfe165d","schema_version":"1.0","event_id":"sha256:abebdad18eedceab79c7322d2872572e6e91bf6565e9ac6bb8e0e1caecfe165d"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2011:WMJ2WD2AU7AXIPPOCOG35J6QPK","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On volumes determined by subsets of Euclidean space","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO","math.MG"],"primary_cat":"math.CA","authors_text":"Alex Iosevich, Allan Greenleaf, Mihalis Mourgoglou","submitted_at":"2011-10-31T13:39:21Z","abstract_excerpt":"Given $E \\subset {\\Bbb R}^d$, define the \\emph{volume set} of $E$, ${\\mathcal V}(E)= \\{det(x^1, x^2, ... x^d): x^j \\in E\\}$. In $\\R^3$, we prove that ${\\mathcal V}(E)$ has positive Lebesgue measure if either the Hausdorff dimension of $E\\subset \\Bbb R^3$ is greater than 13/5, or $E$ is a product set of the form $E=B_1\\times B_2\\times B_3$ with $B_j\\subset\\R,\\, dim_{\\mathcal H}(B_j)>2/3,\\, j=1,2,3$. We show that the same conclusion holds for $\\V(E)$ of Salem subsets $E\\subset\\R^d$ with $\\hde>d-1$, and give applications to discrete combinatorial geometry."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1110.6790","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":"2026-05-18T04:09:54Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ECyrQifP1SspY/vh+XPaAnOgiF/KIbGwror9t0MlPU/bxi0nV63qrDSLoRB3Powg4YgZYp0WE6ihloqu4aObDQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-03T18:36:56.009755Z"},"content_sha256":"af547edefd957cce00950d2fe83f16d9095225aae256d760f051f677b1a5f5f6","schema_version":"1.0","event_id":"sha256:af547edefd957cce00950d2fe83f16d9095225aae256d760f051f677b1a5f5f6"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/WMJ2WD2AU7AXIPPOCOG35J6QPK/bundle.json","state_url":"https://pith.science/pith/WMJ2WD2AU7AXIPPOCOG35J6QPK/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/WMJ2WD2AU7AXIPPOCOG35J6QPK/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":"2026-06-03T18:36:56Z","links":{"resolver":"https://pith.science/pith/WMJ2WD2AU7AXIPPOCOG35J6QPK","bundle":"https://pith.science/pith/WMJ2WD2AU7AXIPPOCOG35J6QPK/bundle.json","state":"https://pith.science/pith/WMJ2WD2AU7AXIPPOCOG35J6QPK/state.json","well_known_bundle":"https://pith.science/.well-known/pith/WMJ2WD2AU7AXIPPOCOG35J6QPK/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:WMJ2WD2AU7AXIPPOCOG35J6QPK","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":"f646ba8dd68d8f0edaf5987b13d3db7e509778c49b1176e03bf0f82c261a2e6e","cross_cats_sorted":["math.CO","math.MG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2011-10-31T13:39:21Z","title_canon_sha256":"caa947aa04f898d49c206cad785bc1e2c7807fd5f068ab563b40d74d0281101d"},"schema_version":"1.0","source":{"id":"1110.6790","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1110.6790","created_at":"2026-05-18T04:09:54Z"},{"alias_kind":"arxiv_version","alias_value":"1110.6790v1","created_at":"2026-05-18T04:09:54Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1110.6790","created_at":"2026-05-18T04:09:54Z"},{"alias_kind":"pith_short_12","alias_value":"WMJ2WD2AU7AX","created_at":"2026-05-18T12:26:44Z"},{"alias_kind":"pith_short_16","alias_value":"WMJ2WD2AU7AXIPPO","created_at":"2026-05-18T12:26:44Z"},{"alias_kind":"pith_short_8","alias_value":"WMJ2WD2A","created_at":"2026-05-18T12:26:44Z"}],"graph_snapshots":[{"event_id":"sha256:af547edefd957cce00950d2fe83f16d9095225aae256d760f051f677b1a5f5f6","target":"graph","created_at":"2026-05-18T04:09:54Z","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":"Given $E \\subset {\\Bbb R}^d$, define the \\emph{volume set} of $E$, ${\\mathcal V}(E)= \\{det(x^1, x^2, ... x^d): x^j \\in E\\}$. In $\\R^3$, we prove that ${\\mathcal V}(E)$ has positive Lebesgue measure if either the Hausdorff dimension of $E\\subset \\Bbb R^3$ is greater than 13/5, or $E$ is a product set of the form $E=B_1\\times B_2\\times B_3$ with $B_j\\subset\\R,\\, dim_{\\mathcal H}(B_j)>2/3,\\, j=1,2,3$. We show that the same conclusion holds for $\\V(E)$ of Salem subsets $E\\subset\\R^d$ with $\\hde>d-1$, and give applications to discrete combinatorial geometry.","authors_text":"Alex Iosevich, Allan Greenleaf, Mihalis Mourgoglou","cross_cats":["math.CO","math.MG"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2011-10-31T13:39:21Z","title":"On volumes determined by subsets of Euclidean space"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1110.6790","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:abebdad18eedceab79c7322d2872572e6e91bf6565e9ac6bb8e0e1caecfe165d","target":"record","created_at":"2026-05-18T04:09:54Z","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":"f646ba8dd68d8f0edaf5987b13d3db7e509778c49b1176e03bf0f82c261a2e6e","cross_cats_sorted":["math.CO","math.MG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2011-10-31T13:39:21Z","title_canon_sha256":"caa947aa04f898d49c206cad785bc1e2c7807fd5f068ab563b40d74d0281101d"},"schema_version":"1.0","source":{"id":"1110.6790","kind":"arxiv","version":1}},"canonical_sha256":"b313ab0f40a7c1743dee138dbea7d07aadb9240f1b1f90f25a5367f1aa650f4a","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"b313ab0f40a7c1743dee138dbea7d07aadb9240f1b1f90f25a5367f1aa650f4a","first_computed_at":"2026-05-18T04:09:54.192749Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:09:54.192749Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"TXCV7kkyRbQxQHgUEXnPFrzIA/YyIe/dyxHccXzdDWLWdbeHXWYgaeUQ9zf9v/hMgSC91sdzUTV/a8f7TGO5Cw==","signature_status":"signed_v1","signed_at":"2026-05-18T04:09:54.193361Z","signed_message":"canonical_sha256_bytes"},"source_id":"1110.6790","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:abebdad18eedceab79c7322d2872572e6e91bf6565e9ac6bb8e0e1caecfe165d","sha256:af547edefd957cce00950d2fe83f16d9095225aae256d760f051f677b1a5f5f6"],"state_sha256":"7d86a3a666c15a0365d741d3b24f6faa3429e81297df3275c8d8a7674743e2a7"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"JqWaHHs3MdD/PHsolSsSRGFQF3P4k8kpUmknC/j18i3YrI3a8GLnwjNph1Kc5oI5iAYKuBT3ldZtLiuiiUT6BQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-03T18:36:56.011774Z","bundle_sha256":"4987ce6e2d92e3237a1a1c31e91a6b842936143e2171593f34b6a76dd483458c"}}