{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2017:HCUZUPA4LUGNSQIZXGK2ESR57K","short_pith_number":"pith:HCUZUPA4","schema_version":"1.0","canonical_sha256":"38a99a3c1c5d0cd94119b995a24a3dfaa8c39dd8644391c0d1ed778b5f9b3df7","source":{"kind":"arxiv","id":"1706.05969","version":5},"attestation_state":"computed","paper":{"title":"Popular differences and generalized Sidon sets","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.NT","authors_text":"Max Wenqiang Xu","submitted_at":"2017-06-19T14:13:16Z","abstract_excerpt":"For a subset $A \\subseteq [N]$, we define the representation function $ r_{A-A}(d) := \\#\\{(a,a') \\in A \\times A : d = a - a'\\}$ and define $M_D(A) := \\max_{1 \\leq d < D} r_{A-A}(d)$ for $D>1$. We study the smallest possible value of $M_D(A)$ as $A$ ranges over all possible subsets of $[N]$ with a given size. We give explicit asymptotic expressions with constant coefficients determined for a large range of $D$. We shall also see how this problem connects to a well-known problem about generalized Sidon sets."},"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":"1706.05969","kind":"arxiv","version":5},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NT","submitted_at":"2017-06-19T14:13:16Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"0c2c4924f0c0afc5db3db72547518d22cd0f5513273a4c3f21cbbd8f123084ed","abstract_canon_sha256":"0fb8a0c3e166f08fa5f9a85f01c7cc7466cc736fe28735261a19154593c66d66"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T00:44:35.389759Z","signature_b64":"EONC6eiLNKYYgczLz/HvXY2BSJWyREu6w8xxvJ6LGVMySz+g40j+msyFuRamsY8rldKQZKNPkN/kd85C48ZoBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"38a99a3c1c5d0cd94119b995a24a3dfaa8c39dd8644391c0d1ed778b5f9b3df7","last_reissued_at":"2026-07-05T00:44:35.389378Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T00:44:35.389378Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Popular differences and generalized Sidon sets","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.NT","authors_text":"Max Wenqiang Xu","submitted_at":"2017-06-19T14:13:16Z","abstract_excerpt":"For a subset $A \\subseteq [N]$, we define the representation function $ r_{A-A}(d) := \\#\\{(a,a') \\in A \\times A : d = a - a'\\}$ and define $M_D(A) := \\max_{1 \\leq d < D} r_{A-A}(d)$ for $D>1$. We study the smallest possible value of $M_D(A)$ as $A$ ranges over all possible subsets of $[N]$ with a given size. We give explicit asymptotic expressions with constant coefficients determined for a large range of $D$. We shall also see how this problem connects to a well-known problem about generalized Sidon sets."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1706.05969","kind":"arxiv","version":5},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/1706.05969/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"1706.05969","created_at":"2026-07-05T00:44:35.389442+00:00"},{"alias_kind":"arxiv_version","alias_value":"1706.05969v5","created_at":"2026-07-05T00:44:35.389442+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1706.05969","created_at":"2026-07-05T00:44:35.389442+00:00"},{"alias_kind":"pith_short_12","alias_value":"HCUZUPA4LUGN","created_at":"2026-07-05T00:44:35.389442+00:00"},{"alias_kind":"pith_short_16","alias_value":"HCUZUPA4LUGNSQIZ","created_at":"2026-07-05T00:44:35.389442+00:00"},{"alias_kind":"pith_short_8","alias_value":"HCUZUPA4","created_at":"2026-07-05T00:44:35.389442+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/HCUZUPA4LUGNSQIZXGK2ESR57K","json":"https://pith.science/pith/HCUZUPA4LUGNSQIZXGK2ESR57K.json","graph_json":"https://pith.science/api/pith-number/HCUZUPA4LUGNSQIZXGK2ESR57K/graph.json","events_json":"https://pith.science/api/pith-number/HCUZUPA4LUGNSQIZXGK2ESR57K/events.json","paper":"https://pith.science/paper/HCUZUPA4"},"agent_actions":{"view_html":"https://pith.science/pith/HCUZUPA4LUGNSQIZXGK2ESR57K","download_json":"https://pith.science/pith/HCUZUPA4LUGNSQIZXGK2ESR57K.json","view_paper":"https://pith.science/paper/HCUZUPA4","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1706.05969&json=true","fetch_graph":"https://pith.science/api/pith-number/HCUZUPA4LUGNSQIZXGK2ESR57K/graph.json","fetch_events":"https://pith.science/api/pith-number/HCUZUPA4LUGNSQIZXGK2ESR57K/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/HCUZUPA4LUGNSQIZXGK2ESR57K/action/timestamp_anchor","attest_storage":"https://pith.science/pith/HCUZUPA4LUGNSQIZXGK2ESR57K/action/storage_attestation","attest_author":"https://pith.science/pith/HCUZUPA4LUGNSQIZXGK2ESR57K/action/author_attestation","sign_citation":"https://pith.science/pith/HCUZUPA4LUGNSQIZXGK2ESR57K/action/citation_signature","submit_replication":"https://pith.science/pith/HCUZUPA4LUGNSQIZXGK2ESR57K/action/replication_record"}},"created_at":"2026-07-05T00:44:35.389442+00:00","updated_at":"2026-07-05T00:44:35.389442+00:00"}