{"paper":{"title":"The Entropic Sum-Product Phenomenon","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.CO","math.IT","math.PR"],"primary_cat":"cs.IT","authors_text":"Rupert Li","submitted_at":"2026-07-31T05:43:56Z","abstract_excerpt":"Let $X,X'$ be independent and identically distributed discrete real-valued random variables of finite Shannon entropy, and write $H(X)$ for the Shannon entropy of $X$. We prove that \\[\n  \\max\\{H(X+X'),\\,H(XX')\\} \\ge \\frac87 H(X)-O(\\log H(X)). \\] This is the entropic analog of the celebrated sum-product phenomenon, and answers a question of Goh, which simply asked for a coefficient strictly larger than 1. An example by the author, Gavalakis, and Kontoyiannis showed the coefficient cannot exceed $\\frac43$. Previous work by Gavalakis, Goh, and Kontoyiannis was able to prove a result of a weaker f"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.29042","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/2607.29042/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"}