Super-Linear Growth of the Capacity-Achieving Input Support for the Amplitude-Constrained AWGN Channel

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.