Various lower bounds are established for the entropy of sums, products and their combinations. First, we derive a prime-field analogue of a version of the entropy power inequality established by Tao over torsion-free groups. Next, we prove an entropy sum-product statement: For independent and identically distributed random variables $X,X'$, the maximum of ${\bf H}(X+X')$ and ${\bf H}(XX')$ is bounded below by a linear combination of the entropy and the min-entropy (Rényi entropy of order~$\infty$) of $X$. This result, obtained by bounding entropies of the form ${\bf H}\bigl( X(Y+Z)\bigr)$ from above and below, is valid over arbitrary fields $F$. Over $F={\bf R}$, a slightly stronger inequality is derived. Finally, a weak version of a purely Shannon-entropic sum-product result is developed: If the entropic additive doubling of a random variable $X$ over an arbitrary field is $O(1)$, then its multiplicative doubling is at least proportional to ${\bf H}(X)$.

翻译：本文建立了和的熵、积的熵及其组合形式的多种下界。首先，我们推导了Tao在无挠群上建立的熵功率不等式的一个版本在素域上的类比。其次，我们证明了一个熵和积定理：对于独立同分布的随机变量$X,X'$，${\bf H}(X+X')$和${\bf H}(XX')$的最大值由$X$的熵与最小熵（Rényi熵，阶数为$\infty$）的线性组合给出下界。这一结果通过从上下界同时约束${\bf H}\bigl( X(Y+Z)\bigr)$形式的熵获得，适用于任意域$F$。当$F={\bf R}$时，我们推导出一个略强的不等式。最后，我们发展了一个纯香农熵和积结果的弱版本：若随机变量$X$在任意域上的熵加性加倍为$O(1)$，则其乘性加倍至少与${\bf H}(X)$成比例。