2026/07/31 by Rupert Li
Computer Science · Mathematics · #cs.IT #math.CO #math.IT #math.PR
40 pages
arxiv created 2026/07/31 · arxiv updated 2026/08/03
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 max\H(X+X'), H(XX')\ ≥ \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 form, which could not translate to a coefficient strictly larger than 1 because of examples where the min-entropy is significantly smaller than the Shannon entropy. By splitting the distribution of X into uniform pieces, which costs O(log H(X)) entropy, we obviate this issue, establishing a coefficient of (10)/(9). We augment this to \frac87 by adapting the work of Solymosi, which established the combinatorial sum-product phenomenon with coefficient \frac43 by bounding the multiplicative energy, to the entropy setting, again via a uniformization technique.