{"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"}