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：针对和、积及其组合的熵，本文建立了多种下界。首先，我们推导了Tao在无挠群上建立的熵幂不等式的一个版本在素数域中的类似结果。接着，我们证明了一个熵和积陈述：对于独立同分布的随机变量$X,X'$，${\bf H}(X+X')$与${\bf H}(XX')$的最大值被$X$的熵与最小熵（Rényi熵，阶数为~$\infty$）的线性组合所下界约束。该结果通过从上下两侧界定形如${\bf H}\bigl( X(Y+Z)\bigr)$的熵获得，适用于任意域$F$。当$F={\bf R}$时，我们推导了一个稍强的不等式。最后，我们发展了一个纯香农熵和积结论的弱版本：若任意域上的随机变量$X$的熵加性倍增为$O(1)$，则其乘性倍增至少与${\bf H}(X)$成正比。