The Entropic Sum-Product Phenomenon

2026-07-31

AI Summary

This work investigates the growth of Shannon entropy for discrete real-valued random variables \(X\) under addition and multiplication, aiming to establish whether there exists a constant \(c > 1\) such that \(\max\{H(X+X'), H(XX')\} \geq c\,H(X)\), where \(X'\) is an independent copy of \(X\). By decomposing the distribution of \(X\) into uniform blocks and combining multiplicative energy estimates with entropy regularization techniques, the authors prove the first sum–product inequality in the Shannon entropy setting with a constant strictly exceeding 1. Specifically, they show that \(\max\{H(X+X'), H(XX')\} \geq \frac{8}{7}H(X) - O(\log H(X))\). This result overcomes obstructions arising from the disparity between min-entropy and Shannon entropy, improving the best-known constant from \(10/9\) to \(8/7\) and significantly advancing toward the conjectured optimal bound of \(4/3\).

Abstract

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 \[ \max\{H(X+X'),\,H(XX')\} \ge \frac87 H(X)-O(\log H(X)). \] 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 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 $\frac{10}{9}$. 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.

Problem

• sum-product phenomenon
• Shannon entropy
• entropic inequality
• random variables
• discrete distributions

Innovation

• entropic sum-product phenomenon
• Shannon entropy
• uniformization technique
• multiplicative energy
• discrete random variables