AI 中文总结
该研究针对和积现象的熵类比问题，通过拆分分布和适配组合和积的相关技术，将原问题的系数提升至8/7，回答了Goh提出的相关问题。
AI 中文摘要
设$X$和$X'$是独立同分布的有限香农熵离散实值随机变量，$H(X)$表示$X$的香农熵。我们证明：$\{H(X+X'),\\,H(XX')\}$的最大值不小于$\frac{8}{7}H(X)-O(\log H(X))$。这是著名和积现象的熵类比，回答了Goh提出的“求严格大于1的系数”的问题。作者、Gavalakis和Kontoyiannis的例子表明该系数不能超过$\frac{4}{3}$。此前Gavalakis、Goh和Kontoyiannis的工作仅能证明较弱形式的结果，因存在最小熵远小于香农熵的例子，无法得到严格大于1的系数。通过将$X$的分布拆分为均匀块（熵损失为$O(\log H(X))$），我们解决了该问题，得到系数$\frac{10}{9}$；通过将Solymosi在组合和积现象中利用乘法能量界得到系数$\frac{4}{3}$的工作适配到熵框架（同样通过均匀化技术），我们将系数提升至$\frac{8}{7}$。
英文摘要
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.
Comments40 pages
