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We show that, up to finitely many exceptions, non-zero sums $x_1+y_1$ and $x_2+y_2$, with $x_1, x_2\\in \\Gamma$ and $y_1,y_2\\in \\Delta$, are multiplicatively dependent only if $x_1/x_2=y_1/y_2$ is a root of unity. For $m\\ge 3$, we discuss possible shapes of $m$ multiplicatively dependent sums $x_1+y_1, \\ \\ldots, \\ x_m+y_m$ with $x_1, \\ldots, x_m \\in \\Gamma$ and $y_1, \\ldots, y_m \\in \\Delta$. For $m=3$ we classify such sums, up to finitely many exceptions, as"},"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":"2607.28857","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2026-07-30T21:44:54Z","cross_cats_sorted":[],"title_canon_sha256":"ccd9a3131d8e51d36f0a40dd9c408d726b2fed01ea7ea3c4de155404e660414e","abstract_canon_sha256":"efce3effb53b4ade4ca9dc5c3ca51d88116e5a482ad4a7155ece63c2f1b806d5"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-08-03T01:17:13.453964Z","signature_b64":"82YqekMRwyuTMVemf8lB/44fl3LExO81rgsALcKdOOX9pMYeQQlNHiEFdWlbakWJ6iIvS8r+IVgrislZXghhBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"fa3429537b7032a487aa2233e50b984ed1ec23218b07fed36024abecf1d43ef6","last_reissued_at":"2026-08-03T01:17:13.451949Z","signature_status":"signed_v1","first_computed_at":"2026-08-03T01:17:13.451949Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Multiplicative dependence in the sumset of multiplicative groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Florian Luca, Yuri Bilu","submitted_at":"2026-07-30T21:44:54Z","abstract_excerpt":"Let $\\Gamma$ and $\\Delta$ be finitely generated multiplicative groups of algebraic numbers such that $\\Gamma\\cap\\Delta$ is a finite group. We show that, up to finitely many exceptions, non-zero sums $x_1+y_1$ and $x_2+y_2$, with $x_1, x_2\\in \\Gamma$ and $y_1,y_2\\in \\Delta$, are multiplicatively dependent only if $x_1/x_2=y_1/y_2$ is a root of unity. For $m\\ge 3$, we discuss possible shapes of $m$ multiplicatively dependent sums $x_1+y_1, \\ \\ldots, \\ x_m+y_m$ with $x_1, \\ldots, x_m \\in \\Gamma$ and $y_1, \\ldots, y_m \\in \\Delta$. For $m=3$ we classify such sums, up to finitely many exceptions, as"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.28857","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/2607.28857/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":"2607.28857","created_at":"2026-08-03T01:17:13.453076+00:00"},{"alias_kind":"arxiv_version","alias_value":"2607.28857v1","created_at":"2026-08-03T01:17:13.453076+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2607.28857","created_at":"2026-08-03T01:17:13.453076+00:00"},{"alias_kind":"pith_short_12","alias_value":"7I2CSU33OAZK","created_at":"2026-08-03T01:17:13.453076+00:00"},{"alias_kind":"pith_short_16","alias_value":"7I2CSU33OAZKJB5K","created_at":"2026-08-03T01:17:13.453076+00:00"},{"alias_kind":"pith_short_8","alias_value":"7I2CSU33","created_at":"2026-08-03T01:17:13.453076+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/7I2CSU33OAZKJB5KEIZ6KC4YJ3","json":"https://pith.science/pith/7I2CSU33OAZKJB5KEIZ6KC4YJ3.json","graph_json":"https://pith.science/api/pith-number/7I2CSU33OAZKJB5KEIZ6KC4YJ3/graph.json","events_json":"https://pith.science/api/pith-number/7I2CSU33OAZKJB5KEIZ6KC4YJ3/events.json","paper":"https://pith.science/paper/7I2CSU33"},"agent_actions":{"view_html":"https://pith.science/pith/7I2CSU33OAZKJB5KEIZ6KC4YJ3","download_json":"https://pith.science/pith/7I2CSU33OAZKJB5KEIZ6KC4YJ3.json","view_paper":"https://pith.science/paper/7I2CSU33","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2607.28857&json=true","fetch_graph":"https://pith.science/api/pith-number/7I2CSU33OAZKJB5KEIZ6KC4YJ3/graph.json","fetch_events":"https://pith.science/api/pith-number/7I2CSU33OAZKJB5KEIZ6KC4YJ3/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/7I2CSU33OAZKJB5KEIZ6KC4YJ3/action/timestamp_anchor","attest_storage":"https://pith.science/pith/7I2CSU33OAZKJB5KEIZ6KC4YJ3/action/storage_attestation","attest_author":"https://pith.science/pith/7I2CSU33OAZKJB5KEIZ6KC4YJ3/action/author_attestation","sign_citation":"https://pith.science/pith/7I2CSU33OAZKJB5KEIZ6KC4YJ3/action/citation_signature","submit_replication":"https://pith.science/pith/7I2CSU33OAZKJB5KEIZ6KC4YJ3/action/replication_record"}},"created_at":"2026-08-03T01:17:13.453076+00:00","updated_at":"2026-08-03T01:17:13.453076+00:00"}