AI 中文总结
本文证明独立离散随机变量和的熵下界达到尖锐常数1/2，推广至无挠阿贝尔群和素数阶循环群，并给出显式误差界，二项分布显示最优性。
AI 中文摘要
对于无挠阿贝尔群和素数阶循环群中同分布的被加数，和的熵的尖锐高熵下界此前已知。对于任意独立的被加数，Gavalakis、Goh 和 Kontoyiannis 获得了 $1/8$ 的加性常数，并猜想尖锐常数为 $1/2$。我们证明，具有有限香农熵的独立离散随机变量 $X,Y$ 满足 $H(X+Y)\ge (H(X)+H(Y))/2+1/2-o(1)$，这在每个无挠阿贝尔群中当 $\max{H(X),H(Y)}\to\infty$ 时成立。同样的结论在素数阶循环群 $\mathbb F_p$ 中也成立，当 $\max{H(X),H(Y)}$ 和 $\log_2 p-\max{H(X),H(Y)}$ 都趋于无穷时。我们在两种情形下都给出了显式误差界。证明提取了一个具有成对点概率的分量，同时独立于其支撑控制余项的熵。离散重排和均匀扰动然后将连续熵幂不等式转移到该分量。在素数阶循环群中，一个额外的估计控制了模约化下损失的熵。二项分布表明常数 $1/2$ 是最优的。
英文摘要
Sharp high-entropy lower bounds for the entropy of a sum were known for identically distributed summands in torsion-free abelian groups and in prime cyclic groups. For arbitrary independent summands, Gavalakis, Goh and Kontoyiannis obtained an additive constant of $1/8$ and conjectured that the sharp constant is $1/2$. We prove that independent discrete random variables $X,Y$ with finite Shannon entropies satisfy $H(X+Y)\ge (H(X)+H(Y))/2+1/2-o(1)$ in every torsion-free abelian group as $\max{H(X),H(Y)}\to\infty$. The same conclusion holds in the prime cyclic group $\mathbb F_p$ when both $\max{H(X),H(Y)}$ and $\log_2 p-\max{H(X),H(Y)}$ tend to infinity. We give explicit error bounds in both settings. The proof extracts a component with paired point probabilities while controlling the entropy of the remainder independently of its support. Discrete rearrangement and uniform perturbation then transfer the continuous entropy power inequality to this component. In prime cyclic groups, an additional estimate controls the entropy lost under modular reduction. Binomial distributions show that the constant $1/2$ is optimal.
