{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:BTXZ752YIXFGXBJQQCAIQCGUBL","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":"ef7ae852c882348142459a8344f6fda97c17c012726f75495c6e56199e7e286b","cross_cats_sorted":["math.MG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-12-15T06:19:57Z","title_canon_sha256":"7d12d25995c3e0b201c5656d889136db32eae4202f78ad06e86765c69d935f41"},"schema_version":"1.0","source":{"id":"1612.04940","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1612.04940","created_at":"2026-05-18T00:54:53Z"},{"alias_kind":"arxiv_version","alias_value":"1612.04940v1","created_at":"2026-05-18T00:54:53Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1612.04940","created_at":"2026-05-18T00:54:53Z"},{"alias_kind":"pith_short_12","alias_value":"BTXZ752YIXFG","created_at":"2026-05-18T12:30:09Z"},{"alias_kind":"pith_short_16","alias_value":"BTXZ752YIXFGXBJQ","created_at":"2026-05-18T12:30:09Z"},{"alias_kind":"pith_short_8","alias_value":"BTXZ752Y","created_at":"2026-05-18T12:30:09Z"}],"graph_snapshots":[{"event_id":"sha256:993d0ae4918554d019947773fb204faef8b8312219a9f29cba5ba4a87ba4c1d6","target":"graph","created_at":"2026-05-18T00:54:53Z","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":"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","authors_text":"Ariel Bruner, Micha Sharir","cross_cats":["math.MG"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-12-15T06:19:57Z","title":"Distinct distances between a collinear set and an arbitrary set of points"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1612.04940","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":"sha256:20d5c7df889f2bda259110f509f80d77c92f6cfe5251418eb945071ddf96d601","target":"record","created_at":"2026-05-18T00:54:53Z","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":"ef7ae852c882348142459a8344f6fda97c17c012726f75495c6e56199e7e286b","cross_cats_sorted":["math.MG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-12-15T06:19:57Z","title_canon_sha256":"7d12d25995c3e0b201c5656d889136db32eae4202f78ad06e86765c69d935f41"},"schema_version":"1.0","source":{"id":"1612.04940","kind":"arxiv","version":1}},"canonical_sha256":"0cef9ff75845ca6b853080808808d40ad0e220a1927250680c6f8f23896e879c","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"0cef9ff75845ca6b853080808808d40ad0e220a1927250680c6f8f23896e879c","first_computed_at":"2026-05-18T00:54:53.855824Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:54:53.855824Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"nRCoNi5pPdS4ze22Hy76jpYGCUSANLriNVKlHmoeaxdOgqFmUAHbYKD7yoYlD4c6cWCAOI3ZPNssQ/ECSbnFCg==","signature_status":"signed_v1","signed_at":"2026-05-18T00:54:53.856663Z","signed_message":"canonical_sha256_bytes"},"source_id":"1612.04940","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:20d5c7df889f2bda259110f509f80d77c92f6cfe5251418eb945071ddf96d601","sha256:993d0ae4918554d019947773fb204faef8b8312219a9f29cba5ba4a87ba4c1d6"],"state_sha256":"0509c9edeb8c8e7747af7256fb0c107a7289754460b5e66861e719208b49bc65"}