{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2017:HCUZUPA4LUGNSQIZXGK2ESR57K","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":"0fb8a0c3e166f08fa5f9a85f01c7cc7466cc736fe28735261a19154593c66d66","cross_cats_sorted":["math.CO"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NT","submitted_at":"2017-06-19T14:13:16Z","title_canon_sha256":"0c2c4924f0c0afc5db3db72547518d22cd0f5513273a4c3f21cbbd8f123084ed"},"schema_version":"1.0","source":{"id":"1706.05969","kind":"arxiv","version":5}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1706.05969","created_at":"2026-07-05T00:44:35Z"},{"alias_kind":"arxiv_version","alias_value":"1706.05969v5","created_at":"2026-07-05T00:44:35Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1706.05969","created_at":"2026-07-05T00:44:35Z"},{"alias_kind":"pith_short_12","alias_value":"HCUZUPA4LUGN","created_at":"2026-07-05T00:44:35Z"},{"alias_kind":"pith_short_16","alias_value":"HCUZUPA4LUGNSQIZ","created_at":"2026-07-05T00:44:35Z"},{"alias_kind":"pith_short_8","alias_value":"HCUZUPA4","created_at":"2026-07-05T00:44:35Z"}],"graph_snapshots":[{"event_id":"sha256:9dc84f2e2df7cc41628069bce5cc613ae58a842b1906112a832e2d75e53237ad","target":"graph","created_at":"2026-07-05T00:44:35Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/1706.05969/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"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.","authors_text":"Max Wenqiang Xu","cross_cats":["math.CO"],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NT","submitted_at":"2017-06-19T14:13:16Z","title":"Popular differences and generalized Sidon sets"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1706.05969","kind":"arxiv","version":5},"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":"sha256:87fc7215234a4296be66ab16101045470192e71658764667b8a252cb81df3a71","target":"record","created_at":"2026-07-05T00:44:35Z","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":"0fb8a0c3e166f08fa5f9a85f01c7cc7466cc736fe28735261a19154593c66d66","cross_cats_sorted":["math.CO"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NT","submitted_at":"2017-06-19T14:13:16Z","title_canon_sha256":"0c2c4924f0c0afc5db3db72547518d22cd0f5513273a4c3f21cbbd8f123084ed"},"schema_version":"1.0","source":{"id":"1706.05969","kind":"arxiv","version":5}},"canonical_sha256":"38a99a3c1c5d0cd94119b995a24a3dfaa8c39dd8644391c0d1ed778b5f9b3df7","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"38a99a3c1c5d0cd94119b995a24a3dfaa8c39dd8644391c0d1ed778b5f9b3df7","first_computed_at":"2026-07-05T00:44:35.389378Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T00:44:35.389378Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"EONC6eiLNKYYgczLz/HvXY2BSJWyREu6w8xxvJ6LGVMySz+g40j+msyFuRamsY8rldKQZKNPkN/kd85C48ZoBg==","signature_status":"signed_v1","signed_at":"2026-07-05T00:44:35.389759Z","signed_message":"canonical_sha256_bytes"},"source_id":"1706.05969","source_kind":"arxiv","source_version":5}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:87fc7215234a4296be66ab16101045470192e71658764667b8a252cb81df3a71","sha256:9dc84f2e2df7cc41628069bce5cc613ae58a842b1906112a832e2d75e53237ad"],"state_sha256":"b66a7f214ca471fc06042f2d80ba79a955f679bb7cafb1cbcb6496f37e5a4e8d"}