{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2016:NL4EYSW4E46ONKGV72VJ7EU5YW","short_pith_number":"pith:NL4EYSW4","canonical_record":{"source":{"id":"1611.05251","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2016-11-16T12:27:33Z","cross_cats_sorted":["math.NT"],"title_canon_sha256":"bb98d06330633a4ff58100474949ddb079debbfcbb6113b1851c4820e318bee4","abstract_canon_sha256":"2501456eb5eceb7c0a7e9344294bf35bba078ddd295645fd6ae866d010ac3e02"},"schema_version":"1.0"},"canonical_sha256":"6af84c4adc273ce6a8d5feaa9f929dc58a4556e0409ac6722dcfab4d282c73c4","source":{"kind":"arxiv","id":"1611.05251","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1611.05251","created_at":"2026-05-18T00:58:27Z"},{"alias_kind":"arxiv_version","alias_value":"1611.05251v1","created_at":"2026-05-18T00:58:27Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1611.05251","created_at":"2026-05-18T00:58:27Z"},{"alias_kind":"pith_short_12","alias_value":"NL4EYSW4E46O","created_at":"2026-05-18T12:30:32Z"},{"alias_kind":"pith_short_16","alias_value":"NL4EYSW4E46ONKGV","created_at":"2026-05-18T12:30:32Z"},{"alias_kind":"pith_short_8","alias_value":"NL4EYSW4","created_at":"2026-05-18T12:30:32Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2016:NL4EYSW4E46ONKGV72VJ7EU5YW","target":"record","payload":{"canonical_record":{"source":{"id":"1611.05251","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2016-11-16T12:27:33Z","cross_cats_sorted":["math.NT"],"title_canon_sha256":"bb98d06330633a4ff58100474949ddb079debbfcbb6113b1851c4820e318bee4","abstract_canon_sha256":"2501456eb5eceb7c0a7e9344294bf35bba078ddd295645fd6ae866d010ac3e02"},"schema_version":"1.0"},"canonical_sha256":"6af84c4adc273ce6a8d5feaa9f929dc58a4556e0409ac6722dcfab4d282c73c4","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:58:27.521258Z","signature_b64":"aKvPJ5cjf9G56vV7z27PB/uYrPFNyiTgDDr6QygdaNhYvB7kExHL34d0web+TMsDgIsH/mSx+AA2C7BGToctCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"6af84c4adc273ce6a8d5feaa9f929dc58a4556e0409ac6722dcfab4d282c73c4","last_reissued_at":"2026-05-18T00:58:27.520662Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:58:27.520662Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1611.05251","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:58:27Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"gH4nohx4uv7sbu0NRyCsyx0/QfWlt0tmBf8daYrdT83nY0LdTUppq674+2ACmh01YwzPXTUm5w9LGrH1IXW6BQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-05T12:15:53.317539Z"},"content_sha256":"5e2faf20ab34af8368d68f50ba1a73488749691cddc2cb8cd2601fde403fab3d","schema_version":"1.0","event_id":"sha256:5e2faf20ab34af8368d68f50ba1a73488749691cddc2cb8cd2601fde403fab3d"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2016:NL4EYSW4E46ONKGV72VJ7EU5YW","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Expanders with superquadratic growth","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.NT"],"primary_cat":"math.CO","authors_text":"Antal Balog, Dmitry Zhelezov, Oliver Roche-Newton","submitted_at":"2016-11-16T12:27:33Z","abstract_excerpt":"We will prove several expanders with exponent strictly greater than $2$. For any finite set $A \\subset \\mathbb R$, we prove the following six-variable expander results: \\begin{align*} |(A-A)(A-A)(A-A)| &\\gg \\frac{|A|^{2+\\frac{1}{8}}}{\\log^{\\frac{17}{16}}|A|}, \\\\ \\left|\\frac{A+A}{A+A}+\\frac{A}{A}\\right| &\\gg \\frac{|A|^{2+\\frac{2}{17}}}{\\log^{\\frac{16}{17}}|A|}, \\\\ \\left|\\frac{AA+AA}{A+A}\\right| &\\gg \\frac{|A|^{2+\\frac{1}{8}}}{\\log |A|}, \\\\ \\left|\\frac{AA+A}{AA+A}\\right| &\\gg \\frac{|A|^{2+\\frac{1}{8}}}{\\log |A|}. \\end{align*}"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1611.05251","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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:58:27Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"oaeZ7CkFTmqig8SnSw5ZJLCp9wdaYD8d8xJ6V7KgJDKCRyQ79wGFQCpxUgZZcVvRCkbY71zTSjIU3vouCHU8Dg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-05T12:15:53.317891Z"},"content_sha256":"907f986da68774ff865a5c268091e82c5636df31d887943480ebc8cdc29ffbce","schema_version":"1.0","event_id":"sha256:907f986da68774ff865a5c268091e82c5636df31d887943480ebc8cdc29ffbce"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/NL4EYSW4E46ONKGV72VJ7EU5YW/bundle.json","state_url":"https://pith.science/pith/NL4EYSW4E46ONKGV72VJ7EU5YW/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/NL4EYSW4E46ONKGV72VJ7EU5YW/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":"2026-06-05T12:15:53Z","links":{"resolver":"https://pith.science/pith/NL4EYSW4E46ONKGV72VJ7EU5YW","bundle":"https://pith.science/pith/NL4EYSW4E46ONKGV72VJ7EU5YW/bundle.json","state":"https://pith.science/pith/NL4EYSW4E46ONKGV72VJ7EU5YW/state.json","well_known_bundle":"https://pith.science/.well-known/pith/NL4EYSW4E46ONKGV72VJ7EU5YW/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:NL4EYSW4E46ONKGV72VJ7EU5YW","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":"2501456eb5eceb7c0a7e9344294bf35bba078ddd295645fd6ae866d010ac3e02","cross_cats_sorted":["math.NT"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2016-11-16T12:27:33Z","title_canon_sha256":"bb98d06330633a4ff58100474949ddb079debbfcbb6113b1851c4820e318bee4"},"schema_version":"1.0","source":{"id":"1611.05251","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1611.05251","created_at":"2026-05-18T00:58:27Z"},{"alias_kind":"arxiv_version","alias_value":"1611.05251v1","created_at":"2026-05-18T00:58:27Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1611.05251","created_at":"2026-05-18T00:58:27Z"},{"alias_kind":"pith_short_12","alias_value":"NL4EYSW4E46O","created_at":"2026-05-18T12:30:32Z"},{"alias_kind":"pith_short_16","alias_value":"NL4EYSW4E46ONKGV","created_at":"2026-05-18T12:30:32Z"},{"alias_kind":"pith_short_8","alias_value":"NL4EYSW4","created_at":"2026-05-18T12:30:32Z"}],"graph_snapshots":[{"event_id":"sha256:907f986da68774ff865a5c268091e82c5636df31d887943480ebc8cdc29ffbce","target":"graph","created_at":"2026-05-18T00:58:27Z","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 will prove several expanders with exponent strictly greater than $2$. For any finite set $A \\subset \\mathbb R$, we prove the following six-variable expander results: \\begin{align*} |(A-A)(A-A)(A-A)| &\\gg \\frac{|A|^{2+\\frac{1}{8}}}{\\log^{\\frac{17}{16}}|A|}, \\\\ \\left|\\frac{A+A}{A+A}+\\frac{A}{A}\\right| &\\gg \\frac{|A|^{2+\\frac{2}{17}}}{\\log^{\\frac{16}{17}}|A|}, \\\\ \\left|\\frac{AA+AA}{A+A}\\right| &\\gg \\frac{|A|^{2+\\frac{1}{8}}}{\\log |A|}, \\\\ \\left|\\frac{AA+A}{AA+A}\\right| &\\gg \\frac{|A|^{2+\\frac{1}{8}}}{\\log |A|}. \\end{align*}","authors_text":"Antal Balog, Dmitry Zhelezov, Oliver Roche-Newton","cross_cats":["math.NT"],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2016-11-16T12:27:33Z","title":"Expanders with superquadratic growth"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1611.05251","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":"sha256:5e2faf20ab34af8368d68f50ba1a73488749691cddc2cb8cd2601fde403fab3d","target":"record","created_at":"2026-05-18T00:58:27Z","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":"2501456eb5eceb7c0a7e9344294bf35bba078ddd295645fd6ae866d010ac3e02","cross_cats_sorted":["math.NT"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2016-11-16T12:27:33Z","title_canon_sha256":"bb98d06330633a4ff58100474949ddb079debbfcbb6113b1851c4820e318bee4"},"schema_version":"1.0","source":{"id":"1611.05251","kind":"arxiv","version":1}},"canonical_sha256":"6af84c4adc273ce6a8d5feaa9f929dc58a4556e0409ac6722dcfab4d282c73c4","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"6af84c4adc273ce6a8d5feaa9f929dc58a4556e0409ac6722dcfab4d282c73c4","first_computed_at":"2026-05-18T00:58:27.520662Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:58:27.520662Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"aKvPJ5cjf9G56vV7z27PB/uYrPFNyiTgDDr6QygdaNhYvB7kExHL34d0web+TMsDgIsH/mSx+AA2C7BGToctCQ==","signature_status":"signed_v1","signed_at":"2026-05-18T00:58:27.521258Z","signed_message":"canonical_sha256_bytes"},"source_id":"1611.05251","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:5e2faf20ab34af8368d68f50ba1a73488749691cddc2cb8cd2601fde403fab3d","sha256:907f986da68774ff865a5c268091e82c5636df31d887943480ebc8cdc29ffbce"],"state_sha256":"dcd4fc34d69f6fd428585d6c837f8e8b81a0fe4931487603d8a75dda957c43ee"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"TlWJO2kBuxXvqjZ9kX1dOdlMV8fHpVgqk6gglJi53dxJgiURaPthnFIpZHB2HUZjJXHmYd5v+eXQEBAlykp5CA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-05T12:15:53.319917Z","bundle_sha256":"c47be47f1d8c8c681c10522bc65b79fcb1d368e2e3e230577b98bbeecb4647b9"}}