{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2014:5RJ45H6GBZZR5QZYBWSXDEVNNA","short_pith_number":"pith:5RJ45H6G","schema_version":"1.0","canonical_sha256":"ec53ce9fc60e731ec3380da57192ad68174145bb3244428dd406336b75b0f9ff","source":{"kind":"arxiv","id":"1406.0781","version":3},"attestation_state":"computed","paper":{"title":"The smallest sets of points not determined by their X-rays","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.MG","authors_text":"Andreas Alpers, David G. Larman","submitted_at":"2014-06-03T17:04:11Z","abstract_excerpt":"Let $F$ be an $n$-point set in $\\mathbb{K}^d$ with $\\mathbb{K}\\in\\{\\mathbb{R},\\mathbb{Z}\\}$ and $d\\geq 2$. A (discrete) X-ray of $F$ in direction $s$ gives the number of points of $F$ on each line parallel to $s$. We define $\\psi_{\\mathbb{K}^d}(m)$ as the minimum number $n$ for which there exist $m$ directions $s_1,...,s_m$ (pairwise linearly independent and spanning $\\mathbb{R}^d$) such that two $n$-point sets in $\\mathbb{K}^d$ exist that have the same X-rays in these directions. The bound $\\psi_{\\mathbb{Z}^d}(m)\\leq 2^{m-1}$ has been observed many times in the literature. In this note we sho"},"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":"1406.0781","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2014-06-03T17:04:11Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"b4106f1888265bb444ef3a8fbba3988ec51888a7b051505d52039d9fcacfe299","abstract_canon_sha256":"312bc536f5f668a5e4457ca52bc45632b3b1ea2058e8ae0606954fae4b0bb1bb"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:53:09.328882Z","signature_b64":"EGXV+xrAdpigFfujFlJxHWtY1K2yRJhymO8K2EyX6rDMIuncJbf56iNDX7NOYZYeIJdme0+mYqaGsp83etM7Ag==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ec53ce9fc60e731ec3380da57192ad68174145bb3244428dd406336b75b0f9ff","last_reissued_at":"2026-05-18T01:53:09.328398Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:53:09.328398Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The smallest sets of points not determined by their X-rays","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.MG","authors_text":"Andreas Alpers, David G. Larman","submitted_at":"2014-06-03T17:04:11Z","abstract_excerpt":"Let $F$ be an $n$-point set in $\\mathbb{K}^d$ with $\\mathbb{K}\\in\\{\\mathbb{R},\\mathbb{Z}\\}$ and $d\\geq 2$. A (discrete) X-ray of $F$ in direction $s$ gives the number of points of $F$ on each line parallel to $s$. We define $\\psi_{\\mathbb{K}^d}(m)$ as the minimum number $n$ for which there exist $m$ directions $s_1,...,s_m$ (pairwise linearly independent and spanning $\\mathbb{R}^d$) such that two $n$-point sets in $\\mathbb{K}^d$ exist that have the same X-rays in these directions. The bound $\\psi_{\\mathbb{Z}^d}(m)\\leq 2^{m-1}$ has been observed many times in the literature. In this note we sho"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1406.0781","kind":"arxiv","version":3},"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":"1406.0781","created_at":"2026-05-18T01:53:09.328473+00:00"},{"alias_kind":"arxiv_version","alias_value":"1406.0781v3","created_at":"2026-05-18T01:53:09.328473+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1406.0781","created_at":"2026-05-18T01:53:09.328473+00:00"},{"alias_kind":"pith_short_12","alias_value":"5RJ45H6GBZZR","created_at":"2026-05-18T12:28:14.216126+00:00"},{"alias_kind":"pith_short_16","alias_value":"5RJ45H6GBZZR5QZY","created_at":"2026-05-18T12:28:14.216126+00:00"},{"alias_kind":"pith_short_8","alias_value":"5RJ45H6G","created_at":"2026-05-18T12:28:14.216126+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/5RJ45H6GBZZR5QZYBWSXDEVNNA","json":"https://pith.science/pith/5RJ45H6GBZZR5QZYBWSXDEVNNA.json","graph_json":"https://pith.science/api/pith-number/5RJ45H6GBZZR5QZYBWSXDEVNNA/graph.json","events_json":"https://pith.science/api/pith-number/5RJ45H6GBZZR5QZYBWSXDEVNNA/events.json","paper":"https://pith.science/paper/5RJ45H6G"},"agent_actions":{"view_html":"https://pith.science/pith/5RJ45H6GBZZR5QZYBWSXDEVNNA","download_json":"https://pith.science/pith/5RJ45H6GBZZR5QZYBWSXDEVNNA.json","view_paper":"https://pith.science/paper/5RJ45H6G","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1406.0781&json=true","fetch_graph":"https://pith.science/api/pith-number/5RJ45H6GBZZR5QZYBWSXDEVNNA/graph.json","fetch_events":"https://pith.science/api/pith-number/5RJ45H6GBZZR5QZYBWSXDEVNNA/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/5RJ45H6GBZZR5QZYBWSXDEVNNA/action/timestamp_anchor","attest_storage":"https://pith.science/pith/5RJ45H6GBZZR5QZYBWSXDEVNNA/action/storage_attestation","attest_author":"https://pith.science/pith/5RJ45H6GBZZR5QZYBWSXDEVNNA/action/author_attestation","sign_citation":"https://pith.science/pith/5RJ45H6GBZZR5QZYBWSXDEVNNA/action/citation_signature","submit_replication":"https://pith.science/pith/5RJ45H6GBZZR5QZYBWSXDEVNNA/action/replication_record"}},"created_at":"2026-05-18T01:53:09.328473+00:00","updated_at":"2026-05-18T01:53:09.328473+00:00"}