{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2014:C6BSWQPRGO6QVDJ4T4G22RUT7C","short_pith_number":"pith:C6BSWQPR","schema_version":"1.0","canonical_sha256":"17832b41f133bd0a8d3c9f0dad4693f89483b09ded78f187d3c2fa4606ed5e5f","source":{"kind":"arxiv","id":"1406.6029","version":2},"attestation_state":"computed","paper":{"title":"A note on the unit distance problem for planar configurations with Q-independent direction set","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.MG","authors_text":"Jonathan Pakianathan, Mark Herman","submitted_at":"2014-06-23T19:26:02Z","abstract_excerpt":"Let $T(n)$ denote the maximum number of unit distances that a set of $n$ points in the Euclidean plane $\\mathbb{R}^2$ can determine with the additional condition that the distinct unit length directions determined by the configuration must be $\\mathbb{Q}$-independent. This is related to the Erdos unit distance problem but with a simplifying additional assumption on the direction set which holds \"generically\".\n  We show that $T(n+1)-T(n)$ is the Hamming weight of $n$, i.e., the number of nonzero binary coefficients in the binary expansion of $n$, and find a formula for $T(n)$ explicitly. In par"},"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":"1406.6029","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2014-06-23T19:26:02Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"508d73984fe5bc5d582ac779e024fab943ff78d781cd980d94f31aa37df45d77","abstract_canon_sha256":"aef837696d82f9469dc65585bb9c49b519d4c4de07d0690bc300993fc3292ffb"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:49:02.310678Z","signature_b64":"roTJ/QkTOWe+Gl0PpxpNkkt24IxM+cRs4tkmiJW7McFxkrIOB4CzplHlCZSI9LGn1wOOf24GaZCQfWeKzIVtDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"17832b41f133bd0a8d3c9f0dad4693f89483b09ded78f187d3c2fa4606ed5e5f","last_reissued_at":"2026-05-18T02:49:02.310244Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:49:02.310244Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A note on the unit distance problem for planar configurations with Q-independent direction set","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.MG","authors_text":"Jonathan Pakianathan, Mark Herman","submitted_at":"2014-06-23T19:26:02Z","abstract_excerpt":"Let $T(n)$ denote the maximum number of unit distances that a set of $n$ points in the Euclidean plane $\\mathbb{R}^2$ can determine with the additional condition that the distinct unit length directions determined by the configuration must be $\\mathbb{Q}$-independent. This is related to the Erdos unit distance problem but with a simplifying additional assumption on the direction set which holds \"generically\".\n  We show that $T(n+1)-T(n)$ is the Hamming weight of $n$, i.e., the number of nonzero binary coefficients in the binary expansion of $n$, and find a formula for $T(n)$ explicitly. In par"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1406.6029","kind":"arxiv","version":2},"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":"1406.6029","created_at":"2026-05-18T02:49:02.310304+00:00"},{"alias_kind":"arxiv_version","alias_value":"1406.6029v2","created_at":"2026-05-18T02:49:02.310304+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1406.6029","created_at":"2026-05-18T02:49:02.310304+00:00"},{"alias_kind":"pith_short_12","alias_value":"C6BSWQPRGO6Q","created_at":"2026-05-18T12:28:22.404517+00:00"},{"alias_kind":"pith_short_16","alias_value":"C6BSWQPRGO6QVDJ4","created_at":"2026-05-18T12:28:22.404517+00:00"},{"alias_kind":"pith_short_8","alias_value":"C6BSWQPR","created_at":"2026-05-18T12:28:22.404517+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/C6BSWQPRGO6QVDJ4T4G22RUT7C","json":"https://pith.science/pith/C6BSWQPRGO6QVDJ4T4G22RUT7C.json","graph_json":"https://pith.science/api/pith-number/C6BSWQPRGO6QVDJ4T4G22RUT7C/graph.json","events_json":"https://pith.science/api/pith-number/C6BSWQPRGO6QVDJ4T4G22RUT7C/events.json","paper":"https://pith.science/paper/C6BSWQPR"},"agent_actions":{"view_html":"https://pith.science/pith/C6BSWQPRGO6QVDJ4T4G22RUT7C","download_json":"https://pith.science/pith/C6BSWQPRGO6QVDJ4T4G22RUT7C.json","view_paper":"https://pith.science/paper/C6BSWQPR","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1406.6029&json=true","fetch_graph":"https://pith.science/api/pith-number/C6BSWQPRGO6QVDJ4T4G22RUT7C/graph.json","fetch_events":"https://pith.science/api/pith-number/C6BSWQPRGO6QVDJ4T4G22RUT7C/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/C6BSWQPRGO6QVDJ4T4G22RUT7C/action/timestamp_anchor","attest_storage":"https://pith.science/pith/C6BSWQPRGO6QVDJ4T4G22RUT7C/action/storage_attestation","attest_author":"https://pith.science/pith/C6BSWQPRGO6QVDJ4T4G22RUT7C/action/author_attestation","sign_citation":"https://pith.science/pith/C6BSWQPRGO6QVDJ4T4G22RUT7C/action/citation_signature","submit_replication":"https://pith.science/pith/C6BSWQPRGO6QVDJ4T4G22RUT7C/action/replication_record"}},"created_at":"2026-05-18T02:49:02.310304+00:00","updated_at":"2026-05-18T02:49:02.310304+00:00"}