{"paper":{"title":"An Entropy Sumset Inequality and Polynomially Fast Convergence to Shannon Capacity Over All Alphabets","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.CC","math.IT"],"primary_cat":"cs.IT","authors_text":"Ameya Velingker, Venkatesan Guruswami","submitted_at":"2014-11-25T19:55:48Z","abstract_excerpt":"We prove a lower estimate on the increase in entropy when two copies of a conditional random variable $X | Y$, with $X$ supported on $\\mathbb{Z}_q=\\{0,1,\\dots,q-1\\}$ for prime $q$, are summed modulo $q$. Specifically, given two i.i.d copies $(X_1,Y_1)$ and $(X_2,Y_2)$ of a pair of random variables $(X,Y)$, with $X$ taking values in $\\mathbb{Z}_q$, we show \\[ H(X_1 + X_2 \\mid Y_1, Y_2) - H(X|Y) \\ge \\alpha(q) \\cdot H(X|Y) (1-H(X|Y)) \\] for some $\\alpha(q) > 0$, where $H(\\cdot)$ is the normalized (by factor $\\log_2 q$) entropy. Our motivation is an effective analysis of the finite-length behavior"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1411.6993","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":""},"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"}