{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2022:XA7AAF4J7FS3NX6LKPYFXZBSKZ","short_pith_number":"pith:XA7AAF4J","canonical_record":{"source":{"id":"2210.00669","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NT","submitted_at":"2022-10-03T01:11:52Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"a184b5e49b99ffe2c2dee42f6b3cc929734fe53c9ba367cb8ccaad89bc128ccb","abstract_canon_sha256":"8ab0f92291089ab41a13f0f08d5eee56f005e2094f3036af3fd0e6ad6a943dd0"},"schema_version":"1.0"},"canonical_sha256":"b83e001789f965b6dfcb53f05be432565b8763fe6fcf05e503ba9cc4dd08365b","source":{"kind":"arxiv","id":"2210.00669","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2210.00669","created_at":"2026-07-05T05:02:46Z"},{"alias_kind":"arxiv_version","alias_value":"2210.00669v1","created_at":"2026-07-05T05:02:46Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2210.00669","created_at":"2026-07-05T05:02:46Z"},{"alias_kind":"pith_short_12","alias_value":"XA7AAF4J7FS3","created_at":"2026-07-05T05:02:46Z"},{"alias_kind":"pith_short_16","alias_value":"XA7AAF4J7FS3NX6L","created_at":"2026-07-05T05:02:46Z"},{"alias_kind":"pith_short_8","alias_value":"XA7AAF4J","created_at":"2026-07-05T05:02:46Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2022:XA7AAF4J7FS3NX6LKPYFXZBSKZ","target":"record","payload":{"canonical_record":{"source":{"id":"2210.00669","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NT","submitted_at":"2022-10-03T01:11:52Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"a184b5e49b99ffe2c2dee42f6b3cc929734fe53c9ba367cb8ccaad89bc128ccb","abstract_canon_sha256":"8ab0f92291089ab41a13f0f08d5eee56f005e2094f3036af3fd0e6ad6a943dd0"},"schema_version":"1.0"},"canonical_sha256":"b83e001789f965b6dfcb53f05be432565b8763fe6fcf05e503ba9cc4dd08365b","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T05:02:46.457934Z","signature_b64":"gJO/ye6Rm2B7gb9MajQ5DcSa8fqpXfuV8sKe9k6/v+kPcY0MPomBcGwhM2U53nKv4GPuhdXDVaFkEzmbSXB6DQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b83e001789f965b6dfcb53f05be432565b8763fe6fcf05e503ba9cc4dd08365b","last_reissued_at":"2026-07-05T05:02:46.457456Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T05:02:46.457456Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2210.00669","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":"2026-07-05T05:02:46Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"+KqR7MuWqsn2eWTpsTCYb64Pf0W14Dvr9E3qJ+adADMt1CXlFa2NUozftD4RscTSwi8a/kl62TWUB4NLGmOeCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-06T23:28:19.049484Z"},"content_sha256":"be011b4dbc6f23d8983a95aaae34105d59b77566fc63d7257ade48eb59849059","schema_version":"1.0","event_id":"sha256:be011b4dbc6f23d8983a95aaae34105d59b77566fc63d7257ade48eb59849059"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2022:XA7AAF4J7FS3NX6LKPYFXZBSKZ","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Sum and Difference Sets in Generalized Dihedral Groups","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.NT","authors_text":"Andrew Keisling, Astrid Lilly, Guilherme Zeus Dantas e Moura, Justin Cheigh, Matthew Phang, Prakod Ngamlamai, Ruben Ascoli, Ryan Jeong, Steven J. Miller","submitted_at":"2022-10-03T01:11:52Z","abstract_excerpt":"Given a group $G$, we say that a set $A \\subseteq G$ has more sums than differences (MSTD) if $|A+A| > |A-A|$, has more differences than sums (MDTS) if $|A+A| < |A-A|$, or is sum-difference balanced if $|A+A| = |A-A|$. A problem of recent interest has been to understand the frequencies of these type of subsets.\n  The seventh author and Vissuet studied the problem for arbitrary finite groups $G$ and proved that almost all subsets $A\\subseteq G$ are sum-difference balanced as $|G|\\to\\infty$. For the dihedral group $D_{2n}$, they conjectured that of the remaining sets, most are MSTD, i.e., there "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2210.00669","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2210.00669/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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-05T05:02:46Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"eGOOLez911/XbhShkXZ/IPLDnSP12/XPsT6JB9nCwsIhIzSUzxQfSAKnMxJP5vRGl48KvIgaKjzGGzE8znjpAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-06T23:28:19.049865Z"},"content_sha256":"83962c1c49c38616ea7f9d9f1dbe179b7276c7b38770f1edb46d82c143931f91","schema_version":"1.0","event_id":"sha256:83962c1c49c38616ea7f9d9f1dbe179b7276c7b38770f1edb46d82c143931f91"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/XA7AAF4J7FS3NX6LKPYFXZBSKZ/bundle.json","state_url":"https://pith.science/pith/XA7AAF4J7FS3NX6LKPYFXZBSKZ/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/XA7AAF4J7FS3NX6LKPYFXZBSKZ/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":"2026-07-06T23:28:19Z","links":{"resolver":"https://pith.science/pith/XA7AAF4J7FS3NX6LKPYFXZBSKZ","bundle":"https://pith.science/pith/XA7AAF4J7FS3NX6LKPYFXZBSKZ/bundle.json","state":"https://pith.science/pith/XA7AAF4J7FS3NX6LKPYFXZBSKZ/state.json","well_known_bundle":"https://pith.science/.well-known/pith/XA7AAF4J7FS3NX6LKPYFXZBSKZ/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2022:XA7AAF4J7FS3NX6LKPYFXZBSKZ","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":"8ab0f92291089ab41a13f0f08d5eee56f005e2094f3036af3fd0e6ad6a943dd0","cross_cats_sorted":["math.CO"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NT","submitted_at":"2022-10-03T01:11:52Z","title_canon_sha256":"a184b5e49b99ffe2c2dee42f6b3cc929734fe53c9ba367cb8ccaad89bc128ccb"},"schema_version":"1.0","source":{"id":"2210.00669","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2210.00669","created_at":"2026-07-05T05:02:46Z"},{"alias_kind":"arxiv_version","alias_value":"2210.00669v1","created_at":"2026-07-05T05:02:46Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2210.00669","created_at":"2026-07-05T05:02:46Z"},{"alias_kind":"pith_short_12","alias_value":"XA7AAF4J7FS3","created_at":"2026-07-05T05:02:46Z"},{"alias_kind":"pith_short_16","alias_value":"XA7AAF4J7FS3NX6L","created_at":"2026-07-05T05:02:46Z"},{"alias_kind":"pith_short_8","alias_value":"XA7AAF4J","created_at":"2026-07-05T05:02:46Z"}],"graph_snapshots":[{"event_id":"sha256:83962c1c49c38616ea7f9d9f1dbe179b7276c7b38770f1edb46d82c143931f91","target":"graph","created_at":"2026-07-05T05:02:46Z","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/2210.00669/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Given a group $G$, we say that a set $A \\subseteq G$ has more sums than differences (MSTD) if $|A+A| > |A-A|$, has more differences than sums (MDTS) if $|A+A| < |A-A|$, or is sum-difference balanced if $|A+A| = |A-A|$. A problem of recent interest has been to understand the frequencies of these type of subsets.\n  The seventh author and Vissuet studied the problem for arbitrary finite groups $G$ and proved that almost all subsets $A\\subseteq G$ are sum-difference balanced as $|G|\\to\\infty$. For the dihedral group $D_{2n}$, they conjectured that of the remaining sets, most are MSTD, i.e., there ","authors_text":"Andrew Keisling, Astrid Lilly, Guilherme Zeus Dantas e Moura, Justin Cheigh, Matthew Phang, Prakod Ngamlamai, Ruben Ascoli, Ryan Jeong, Steven J. Miller","cross_cats":["math.CO"],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NT","submitted_at":"2022-10-03T01:11:52Z","title":"Sum and Difference Sets in Generalized Dihedral Groups"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2210.00669","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:be011b4dbc6f23d8983a95aaae34105d59b77566fc63d7257ade48eb59849059","target":"record","created_at":"2026-07-05T05:02:46Z","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":"8ab0f92291089ab41a13f0f08d5eee56f005e2094f3036af3fd0e6ad6a943dd0","cross_cats_sorted":["math.CO"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NT","submitted_at":"2022-10-03T01:11:52Z","title_canon_sha256":"a184b5e49b99ffe2c2dee42f6b3cc929734fe53c9ba367cb8ccaad89bc128ccb"},"schema_version":"1.0","source":{"id":"2210.00669","kind":"arxiv","version":1}},"canonical_sha256":"b83e001789f965b6dfcb53f05be432565b8763fe6fcf05e503ba9cc4dd08365b","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"b83e001789f965b6dfcb53f05be432565b8763fe6fcf05e503ba9cc4dd08365b","first_computed_at":"2026-07-05T05:02:46.457456Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T05:02:46.457456Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"gJO/ye6Rm2B7gb9MajQ5DcSa8fqpXfuV8sKe9k6/v+kPcY0MPomBcGwhM2U53nKv4GPuhdXDVaFkEzmbSXB6DQ==","signature_status":"signed_v1","signed_at":"2026-07-05T05:02:46.457934Z","signed_message":"canonical_sha256_bytes"},"source_id":"2210.00669","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:be011b4dbc6f23d8983a95aaae34105d59b77566fc63d7257ade48eb59849059","sha256:83962c1c49c38616ea7f9d9f1dbe179b7276c7b38770f1edb46d82c143931f91"],"state_sha256":"6dac660ad95a78b1432f6ce2fd3d47a4e79e73826404f0f10257b3285b76b2de"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"RSq4ZabqeJRkNaqkfgY7oR4fZjarkjZjxw7MDt1myTlfuII6dX/9XCe1kIaUB/HjsLcVGFioavR1cdLCa5qzBg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-07-06T23:28:19.052029Z","bundle_sha256":"6966ffd4932d2b0cb6c0a2969641b702370d5edd8f17f0094b9fdcd5c6da2621"}}