{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2018:AO3ORGH4UHAD6SIA5VQNASOYP6","short_pith_number":"pith:AO3ORGH4","canonical_record":{"source":{"id":"1805.08900","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-05-22T23:01:43Z","cross_cats_sorted":["math.NT"],"title_canon_sha256":"344093b26e4f79f5ca72e4578a077fd6dd5a77b4d4356c8a1da558ea2fc75d7d","abstract_canon_sha256":"920bf29b76555fbfaccbb8de6b9e1e61bf05b84b8f13f48204b81bf752051ddb"},"schema_version":"1.0"},"canonical_sha256":"03b6e898fca1c03f4900ed60d049d87fbdc3df1254f08fc6e0b1bf8762245a3a","source":{"kind":"arxiv","id":"1805.08900","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1805.08900","created_at":"2026-05-17T23:50:38Z"},{"alias_kind":"arxiv_version","alias_value":"1805.08900v2","created_at":"2026-05-17T23:50:38Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1805.08900","created_at":"2026-05-17T23:50:38Z"},{"alias_kind":"pith_short_12","alias_value":"AO3ORGH4UHAD","created_at":"2026-05-18T12:32:13Z"},{"alias_kind":"pith_short_16","alias_value":"AO3ORGH4UHAD6SIA","created_at":"2026-05-18T12:32:13Z"},{"alias_kind":"pith_short_8","alias_value":"AO3ORGH4","created_at":"2026-05-18T12:32:13Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2018:AO3ORGH4UHAD6SIA5VQNASOYP6","target":"record","payload":{"canonical_record":{"source":{"id":"1805.08900","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-05-22T23:01:43Z","cross_cats_sorted":["math.NT"],"title_canon_sha256":"344093b26e4f79f5ca72e4578a077fd6dd5a77b4d4356c8a1da558ea2fc75d7d","abstract_canon_sha256":"920bf29b76555fbfaccbb8de6b9e1e61bf05b84b8f13f48204b81bf752051ddb"},"schema_version":"1.0"},"canonical_sha256":"03b6e898fca1c03f4900ed60d049d87fbdc3df1254f08fc6e0b1bf8762245a3a","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:50:38.641296Z","signature_b64":"3JoJPVEUsvAFu3hhY7QiC9TlcS2joGUjHhndWNFCR4EU8hH5SLX6pfFeeu1FWuxC2CbUUdVfmMk8Vgn/I5V6Dg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"03b6e898fca1c03f4900ed60d049d87fbdc3df1254f08fc6e0b1bf8762245a3a","last_reissued_at":"2026-05-17T23:50:38.640911Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:50:38.640911Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1805.08900","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-17T23:50:38Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"0alpTk9SqRUxchj42GQQJKENE24Wa3LFwRF+czE2A8X9ZAGSwIARreaYomsN9AbYgxEaaOs7Elc7aRd0COSVAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-02T13:22:59.862261Z"},"content_sha256":"5327b95c2aea13775b97a2de50afb491702ec503f58acb98c128cde9f6333faf","schema_version":"1.0","event_id":"sha256:5327b95c2aea13775b97a2de50afb491702ec503f58acb98c128cde9f6333faf"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2018:AO3ORGH4UHAD6SIA5VQNASOYP6","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"A new bound on Erd\\H{o}s distinct distances problem in the plane over prime fields","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.NT"],"primary_cat":"math.CO","authors_text":"Alex Iosevich, Chun-Yen Shen, Doowon Koh, Le Anh Vinh, Thang Pham","submitted_at":"2018-05-22T23:01:43Z","abstract_excerpt":"In this paper we obtain a new lower bound on the Erd\\H{o}s distinct distances problem in the plane over prime fields. More precisely, we show that for any set $A\\subset \\mathbb{F}_p^2$ with $|A|\\le p^{7/6}$, the number of distinct distances determined by pairs of points in $A$ satisfies $$ |\\Delta(A)| \\gg |A|^{\\frac{1}{2}+\\frac{149}{4214}}.$$ Our result gives a new lower bound of $|\\Delta{(A)}|$ in the range $|A|\\le p^{1+\\frac{149}{4065}}$.\n  The main tools we employ are the energy of a set on a paraboloid due to Rudnev and Shkredov, a point-line incidence bound given by Stevens and de Zeeuw, "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1805.08900","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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-17T23:50:38Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"65xnmO1ewUl0te9TyWaDPerob7L8gHuCJQTrtzMU70jhTQDZpTPxssHUIoUeHYx81Z0jkQ80qpIWMYNpnfliCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-02T13:22:59.862956Z"},"content_sha256":"d84e5e0223ae09009a35c35a49528834d61c437a03b65bd01e68977e2986e8c5","schema_version":"1.0","event_id":"sha256:d84e5e0223ae09009a35c35a49528834d61c437a03b65bd01e68977e2986e8c5"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/AO3ORGH4UHAD6SIA5VQNASOYP6/bundle.json","state_url":"https://pith.science/pith/AO3ORGH4UHAD6SIA5VQNASOYP6/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/AO3ORGH4UHAD6SIA5VQNASOYP6/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-06-02T13:22:59Z","links":{"resolver":"https://pith.science/pith/AO3ORGH4UHAD6SIA5VQNASOYP6","bundle":"https://pith.science/pith/AO3ORGH4UHAD6SIA5VQNASOYP6/bundle.json","state":"https://pith.science/pith/AO3ORGH4UHAD6SIA5VQNASOYP6/state.json","well_known_bundle":"https://pith.science/.well-known/pith/AO3ORGH4UHAD6SIA5VQNASOYP6/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:AO3ORGH4UHAD6SIA5VQNASOYP6","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":"920bf29b76555fbfaccbb8de6b9e1e61bf05b84b8f13f48204b81bf752051ddb","cross_cats_sorted":["math.NT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-05-22T23:01:43Z","title_canon_sha256":"344093b26e4f79f5ca72e4578a077fd6dd5a77b4d4356c8a1da558ea2fc75d7d"},"schema_version":"1.0","source":{"id":"1805.08900","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1805.08900","created_at":"2026-05-17T23:50:38Z"},{"alias_kind":"arxiv_version","alias_value":"1805.08900v2","created_at":"2026-05-17T23:50:38Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1805.08900","created_at":"2026-05-17T23:50:38Z"},{"alias_kind":"pith_short_12","alias_value":"AO3ORGH4UHAD","created_at":"2026-05-18T12:32:13Z"},{"alias_kind":"pith_short_16","alias_value":"AO3ORGH4UHAD6SIA","created_at":"2026-05-18T12:32:13Z"},{"alias_kind":"pith_short_8","alias_value":"AO3ORGH4","created_at":"2026-05-18T12:32:13Z"}],"graph_snapshots":[{"event_id":"sha256:d84e5e0223ae09009a35c35a49528834d61c437a03b65bd01e68977e2986e8c5","target":"graph","created_at":"2026-05-17T23:50:38Z","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":"In this paper we obtain a new lower bound on the Erd\\H{o}s distinct distances problem in the plane over prime fields. More precisely, we show that for any set $A\\subset \\mathbb{F}_p^2$ with $|A|\\le p^{7/6}$, the number of distinct distances determined by pairs of points in $A$ satisfies $$ |\\Delta(A)| \\gg |A|^{\\frac{1}{2}+\\frac{149}{4214}}.$$ Our result gives a new lower bound of $|\\Delta{(A)}|$ in the range $|A|\\le p^{1+\\frac{149}{4065}}$.\n  The main tools we employ are the energy of a set on a paraboloid due to Rudnev and Shkredov, a point-line incidence bound given by Stevens and de Zeeuw, ","authors_text":"Alex Iosevich, Chun-Yen Shen, Doowon Koh, Le Anh Vinh, Thang Pham","cross_cats":["math.NT"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-05-22T23:01:43Z","title":"A new bound on Erd\\H{o}s distinct distances problem in the plane over prime fields"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1805.08900","kind":"arxiv","version":2},"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":"sha256:5327b95c2aea13775b97a2de50afb491702ec503f58acb98c128cde9f6333faf","target":"record","created_at":"2026-05-17T23:50:38Z","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":"920bf29b76555fbfaccbb8de6b9e1e61bf05b84b8f13f48204b81bf752051ddb","cross_cats_sorted":["math.NT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-05-22T23:01:43Z","title_canon_sha256":"344093b26e4f79f5ca72e4578a077fd6dd5a77b4d4356c8a1da558ea2fc75d7d"},"schema_version":"1.0","source":{"id":"1805.08900","kind":"arxiv","version":2}},"canonical_sha256":"03b6e898fca1c03f4900ed60d049d87fbdc3df1254f08fc6e0b1bf8762245a3a","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"03b6e898fca1c03f4900ed60d049d87fbdc3df1254f08fc6e0b1bf8762245a3a","first_computed_at":"2026-05-17T23:50:38.640911Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:50:38.640911Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"3JoJPVEUsvAFu3hhY7QiC9TlcS2joGUjHhndWNFCR4EU8hH5SLX6pfFeeu1FWuxC2CbUUdVfmMk8Vgn/I5V6Dg==","signature_status":"signed_v1","signed_at":"2026-05-17T23:50:38.641296Z","signed_message":"canonical_sha256_bytes"},"source_id":"1805.08900","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:5327b95c2aea13775b97a2de50afb491702ec503f58acb98c128cde9f6333faf","sha256:d84e5e0223ae09009a35c35a49528834d61c437a03b65bd01e68977e2986e8c5"],"state_sha256":"465e7f4c215cfd57fd281fe45ba12cfa3b12bb71175237add6a14a8d11b7cacf"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"4Vs63BFJ2657O1B3JeXH7cDc0cASxlz1HsNz4yi7J5Nf2Uk7pGhF+u6c/7X6wcdDZ7pQ5GX86Z81b1LmN/2KBw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-02T13:22:59.866423Z","bundle_sha256":"a7136f03af86b03fab7fd166aa55ea97829de0f29f722b4a7b86e0b922edda1c"}}