We study the growth of the support size of the capacity-achieving input distribution for the amplitude-constrained additive white Gaussian noise (AWGN) channel. While it is known since Smith (1971) that the optimal input is discrete with finitely many mass points, tight bounds on the number of support points $K(A)$ as the amplitude constraint $A$ increases remain open. Building on recent work by Dytso \emph{et al.} (2019) and Mattingly \emph{et al.} (2018), we derive a new analytical lower bound showing that $K(A)$ grows super-linearly in $A$. Our approach combines total-variation convergence of the output distribution to the uniform law with quantitative limits on Gaussian mixture approximation.

翻译：本文研究幅度受限加性高斯白噪声（AWGN）信道容量可达输入分布的支撑集规模增长特性。自Smith（1971）的研究以来，已知最优输入为具有有限个质量点的离散分布，但随着幅度约束$A$增大，支撑点数量$K(A)$的紧致界仍悬而未决。基于Dytso等人（2019）和Mattingly等人（2018）的最新研究，我们推导出新的解析下界，证明$K(A)$在$A$上呈超线性增长。该方法将输出分布向均匀律的全变差收敛与高斯混合逼近的量化极限相结合。