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The number of distinct distances between $P_1$ and $P_2$ is then $$ \\Omega\\left(\\min\\left\\{ n^{2/3}m^{2/3},\\; \\frac{n^{10/11}m^{4/11}}{\\log^{2/11}m},\\; n^2,\\; m^2\\right\\}\\right) . $$ Without the assumption on $P_2$, there e"},"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":"1612.04940","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-12-15T06:19:57Z","cross_cats_sorted":["math.MG"],"title_canon_sha256":"7d12d25995c3e0b201c5656d889136db32eae4202f78ad06e86765c69d935f41","abstract_canon_sha256":"ef7ae852c882348142459a8344f6fda97c17c012726f75495c6e56199e7e286b"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:54:53.856663Z","signature_b64":"nRCoNi5pPdS4ze22Hy76jpYGCUSANLriNVKlHmoeaxdOgqFmUAHbYKD7yoYlD4c6cWCAOI3ZPNssQ/ECSbnFCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"0cef9ff75845ca6b853080808808d40ad0e220a1927250680c6f8f23896e879c","last_reissued_at":"2026-05-18T00:54:53.855824Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:54:53.855824Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Distinct distances between a collinear set and an arbitrary set of points","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.MG"],"primary_cat":"math.CO","authors_text":"Ariel Bruner, Micha Sharir","submitted_at":"2016-12-15T06:19:57Z","abstract_excerpt":"We consider the number of distinct distances between two finite sets of points in ${\\bf R}^k$, for any constant dimension $k\\ge 2$, where one set $P_1$ consists of $n$ points on a line $l$, and the other set $P_2$ consists of $m$ arbitrary points, such that no hyperplane orthogonal to $l$ and no hypercylinder having $l$ as its axis contains more than $O(1)$ points of $P_2$. The number of distinct distances between $P_1$ and $P_2$ is then $$ \\Omega\\left(\\min\\left\\{ n^{2/3}m^{2/3},\\; \\frac{n^{10/11}m^{4/11}}{\\log^{2/11}m},\\; n^2,\\; m^2\\right\\}\\right) . $$ Without the assumption on $P_2$, there e"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1612.04940","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":"1612.04940","created_at":"2026-05-18T00:54:53.856174+00:00"},{"alias_kind":"arxiv_version","alias_value":"1612.04940v1","created_at":"2026-05-18T00:54:53.856174+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1612.04940","created_at":"2026-05-18T00:54:53.856174+00:00"},{"alias_kind":"pith_short_12","alias_value":"BTXZ752YIXFG","created_at":"2026-05-18T12:30:09.641336+00:00"},{"alias_kind":"pith_short_16","alias_value":"BTXZ752YIXFGXBJQ","created_at":"2026-05-18T12:30:09.641336+00:00"},{"alias_kind":"pith_short_8","alias_value":"BTXZ752Y","created_at":"2026-05-18T12:30:09.641336+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/BTXZ752YIXFGXBJQQCAIQCGUBL","json":"https://pith.science/pith/BTXZ752YIXFGXBJQQCAIQCGUBL.json","graph_json":"https://pith.science/api/pith-number/BTXZ752YIXFGXBJQQCAIQCGUBL/graph.json","events_json":"https://pith.science/api/pith-number/BTXZ752YIXFGXBJQQCAIQCGUBL/events.json","paper":"https://pith.science/paper/BTXZ752Y"},"agent_actions":{"view_html":"https://pith.science/pith/BTXZ752YIXFGXBJQQCAIQCGUBL","download_json":"https://pith.science/pith/BTXZ752YIXFGXBJQQCAIQCGUBL.json","view_paper":"https://pith.science/paper/BTXZ752Y","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1612.04940&json=true","fetch_graph":"https://pith.science/api/pith-number/BTXZ752YIXFGXBJQQCAIQCGUBL/graph.json","fetch_events":"https://pith.science/api/pith-number/BTXZ752YIXFGXBJQQCAIQCGUBL/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/BTXZ752YIXFGXBJQQCAIQCGUBL/action/timestamp_anchor","attest_storage":"https://pith.science/pith/BTXZ752YIXFGXBJQQCAIQCGUBL/action/storage_attestation","attest_author":"https://pith.science/pith/BTXZ752YIXFGXBJQQCAIQCGUBL/action/author_attestation","sign_citation":"https://pith.science/pith/BTXZ752YIXFGXBJQQCAIQCGUBL/action/citation_signature","submit_replication":"https://pith.science/pith/BTXZ752YIXFGXBJQQCAIQCGUBL/action/replication_record"}},"created_at":"2026-05-18T00:54:53.856174+00:00","updated_at":"2026-05-18T00:54:53.856174+00:00"}