{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2006:IBEQ6HQY6JIWG4EC3PEBNEER2G","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":"f51b13ad956dbc18a63f14ca998733658e32f88d0c555484ee4a04a555e8e14a","cross_cats_sorted":["math.CO"],"license":"","primary_cat":"math.NT","submitted_at":"2006-11-19T20:23:47Z","title_canon_sha256":"9d45905f16316c3584a54a40c7a376d975a0e2c3464d0955a4dfd2df6fb99616"},"schema_version":"1.0","source":{"id":"math/0611582","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math/0611582","created_at":"2026-05-18T01:38:23Z"},{"alias_kind":"arxiv_version","alias_value":"math/0611582v2","created_at":"2026-05-18T01:38:23Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0611582","created_at":"2026-05-18T01:38:23Z"},{"alias_kind":"pith_short_12","alias_value":"IBEQ6HQY6JIW","created_at":"2026-05-18T12:25:53Z"},{"alias_kind":"pith_short_16","alias_value":"IBEQ6HQY6JIWG4EC","created_at":"2026-05-18T12:25:53Z"},{"alias_kind":"pith_short_8","alias_value":"IBEQ6HQY","created_at":"2026-05-18T12:25:53Z"}],"graph_snapshots":[{"event_id":"sha256:9c5c000f7702abe768e10f352bdb1897183187f7a5e3ea8ed1e7755e090ab50d","target":"graph","created_at":"2026-05-18T01:38:23Z","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 present a variety of new results on finite sets A of integers for which the sumset A+A is larger than the difference set A-A, so-called MSTD (more sums than differences) sets. First we show that there is, up to affine transformation, a unique MSTD subset of {\\bf Z} of size 8. Secondly, starting from some examples of size 9, we present several new constructions of infinite families of MSTD sets. Thirdly we show that for every fixed ordered pair of non-negative integers (j,k), as n -> \\infty a positive proportion of the subsets of {0,1,2,...,n} satisfy |A+A| = (2n+1) - j, |A-A| = (2n+1) - 2k.","authors_text":"Peter Hegarty","cross_cats":["math.CO"],"headline":"","license":"","primary_cat":"math.NT","submitted_at":"2006-11-19T20:23:47Z","title":"Some explicit constructions of sets with more sums than differences"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0611582","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:ddb2f69a2bb3c917c8495ee2331a092e6cb32b4bcdfd743fba173fe158f402a6","target":"record","created_at":"2026-05-18T01:38:23Z","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":"f51b13ad956dbc18a63f14ca998733658e32f88d0c555484ee4a04a555e8e14a","cross_cats_sorted":["math.CO"],"license":"","primary_cat":"math.NT","submitted_at":"2006-11-19T20:23:47Z","title_canon_sha256":"9d45905f16316c3584a54a40c7a376d975a0e2c3464d0955a4dfd2df6fb99616"},"schema_version":"1.0","source":{"id":"math/0611582","kind":"arxiv","version":2}},"canonical_sha256":"40490f1e18f251637082dbc8169091d18e465667aa6b1f23a5df145da193b7da","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"40490f1e18f251637082dbc8169091d18e465667aa6b1f23a5df145da193b7da","first_computed_at":"2026-05-18T01:38:23.566258Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:38:23.566258Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"iMsBnQ3zKLhorro6I2TCvCtFwnI1qHNW9CWkP8bEQ1FzWCaSxN053fn6AXVvN9vcphKmacTaf1uCEDEnlR9XBA==","signature_status":"signed_v1","signed_at":"2026-05-18T01:38:23.566974Z","signed_message":"canonical_sha256_bytes"},"source_id":"math/0611582","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:ddb2f69a2bb3c917c8495ee2331a092e6cb32b4bcdfd743fba173fe158f402a6","sha256:9c5c000f7702abe768e10f352bdb1897183187f7a5e3ea8ed1e7755e090ab50d"],"state_sha256":"44684a9843c8337c872457c8f89c68eb3c035a756dfd1beb6d8dcfc9dfd5817f"}