Support Size of ε-Capacity-Achieving Inputs for the Amplitude-Constrained AWGN Channel

We study the discrete-time amplitude-constrained additive white Gaussian noise (AWGN) channel from the perspective of near-optimal input distributions in the high-SNR, or equivalently large-amplitude, regime. While it is known that the capacity-achieving input is discrete with finitely many mass points, the precise scaling of its support size as a function of the amplitude constraint remains an open problem. In this work, we instead consider the minimal support size required to achieve capacity up to an ε-gap. We introduce the quantity Kε(A), defined as the smallest support size among discrete inputs supported on [-A,A] that achieves mutual information within ε of capacity. We show that this relaxed formulation is significantly more tractable and admits sharp characterizations in several vanishing-gap regimes. In particular, for polynomially decaying gaps, ε=A-β with β≥1, we establish that Kε(A)=Θ(AlogA) as A→∞. For exponentially small gaps, we obtain bounds of order between AlogA and A3/2. Our approach combines approximation-theoretic bounds for Gaussian mixtures with information-theoretic control of entropy via χ2-divergence, together with a wrapping argument that relates the problem to approximating the uniform distribution on a circle. Beyond the technical results, our framework provides a conceptual explanation for the variety of scaling laws observed in prior numerical studies, suggesting that these may correspond to different regimes of ε-optimality rather than intrinsic properties of the exact optimizer.