{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2011:WMJ2WD2AU7AXIPPOCOG35J6QPK","short_pith_number":"pith:WMJ2WD2A","schema_version":"1.0","canonical_sha256":"b313ab0f40a7c1743dee138dbea7d07aadb9240f1b1f90f25a5367f1aa650f4a","source":{"kind":"arxiv","id":"1110.6790","version":1},"attestation_state":"computed","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."},"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":"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"},"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"},"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1110.6790","created_at":"2026-05-18T04:09:54.192843+00:00"},{"alias_kind":"arxiv_version","alias_value":"1110.6790v1","created_at":"2026-05-18T04:09:54.192843+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1110.6790","created_at":"2026-05-18T04:09:54.192843+00:00"},{"alias_kind":"pith_short_12","alias_value":"WMJ2WD2AU7AX","created_at":"2026-05-18T12:26:44.992195+00:00"},{"alias_kind":"pith_short_16","alias_value":"WMJ2WD2AU7AXIPPO","created_at":"2026-05-18T12:26:44.992195+00:00"},{"alias_kind":"pith_short_8","alias_value":"WMJ2WD2A","created_at":"2026-05-18T12:26:44.992195+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/WMJ2WD2AU7AXIPPOCOG35J6QPK","json":"https://pith.science/pith/WMJ2WD2AU7AXIPPOCOG35J6QPK.json","graph_json":"https://pith.science/api/pith-number/WMJ2WD2AU7AXIPPOCOG35J6QPK/graph.json","events_json":"https://pith.science/api/pith-number/WMJ2WD2AU7AXIPPOCOG35J6QPK/events.json","paper":"https://pith.science/paper/WMJ2WD2A"},"agent_actions":{"view_html":"https://pith.science/pith/WMJ2WD2AU7AXIPPOCOG35J6QPK","download_json":"https://pith.science/pith/WMJ2WD2AU7AXIPPOCOG35J6QPK.json","view_paper":"https://pith.science/paper/WMJ2WD2A","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1110.6790&json=true","fetch_graph":"https://pith.science/api/pith-number/WMJ2WD2AU7AXIPPOCOG35J6QPK/graph.json","fetch_events":"https://pith.science/api/pith-number/WMJ2WD2AU7AXIPPOCOG35J6QPK/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/WMJ2WD2AU7AXIPPOCOG35J6QPK/action/timestamp_anchor","attest_storage":"https://pith.science/pith/WMJ2WD2AU7AXIPPOCOG35J6QPK/action/storage_attestation","attest_author":"https://pith.science/pith/WMJ2WD2AU7AXIPPOCOG35J6QPK/action/author_attestation","sign_citation":"https://pith.science/pith/WMJ2WD2AU7AXIPPOCOG35J6QPK/action/citation_signature","submit_replication":"https://pith.science/pith/WMJ2WD2AU7AXIPPOCOG35J6QPK/action/replication_record"}},"created_at":"2026-05-18T04:09:54.192843+00:00","updated_at":"2026-05-18T04:09:54.192843+00:00"}