We study the amplitude-constrained additive white Gaussian noise (AWGN) channel from the perspective of near-optimal input distributions. 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 $\varepsilon$-gap. We introduce the quantity $K_\varepsilon(A)$, defined as the smallest support size among discrete inputs supported on $[-A,A]$ that achieves mutual information within $\varepsilon$ of capacity. We show that this relaxed formulation is significantly more tractable and admits sharp characterizations across different regimes of $\varepsilon$. In particular, when $\varepsilon$ decays polynomially with $A$, i.e., $\varepsilon = A^{-β}$ for $β\geq 1$, we establish that $K_\varepsilon(A) = Θ(A\sqrt{\log A})$. For exponentially small gaps, we obtain bounds of order between $A\sqrt{\log A}$ and $A^{3/2}$. Our approach combines approximation-theoretic bounds for Gaussian mixtures with information-theoretic control of entropy via $χ^2$-divergence, together with a wrapping argument that relates the problem to approximating the uniform distribution on the circle. Beyond the technical results, our framework provides a conceptual explanation for the variety of scaling laws observed in prior numerical studies, showing that these correspond to different regimes of $\varepsilon$-optimality rather than intrinsic properties of the exact optimizer.

翻译：我们从近优输入分布的角度研究幅度受限加性高斯白噪声（AWGN）信道。虽然已知容量可达输入是离散的且具有有限个质量点，但其支撑集大小随幅度约束变化的精确标度仍是一个开放问题。本文转而考虑在容量$\varepsilon$-差距内所需的最小支撑集大小。我们引入量$K_\varepsilon(A)$，定义为定义在$[-A,A]$上、在$\varepsilon$容量差距内达到互信息的离散输入的最小支撑集大小。我们证明这一松弛形式显著更易处理，并在$\varepsilon$的不同区间内给出精确刻画。特别地，当$\varepsilon$随$A$多项式衰减时（即$\varepsilon = A^{-β}$，$β\geq 1$），我们建立$K_\varepsilon(A) = Θ(A\sqrt{\log A})$。对于指数级小间隙，我们得到$A\sqrt{\log A}$与$A^{3/2}$之间的界。我们的方法将高斯混合的逼近理论界与通过$\chi^2$-散度控制熵的信息论工具相结合，并利用一个将问题关联到圆上均匀分布逼近的环绕论证。除技术结果外，我们的框架为先前数值研究中观察到的多种标度律提供了概念性解释，表明这些标度对应$\varepsilon$-最优性的不同区间，而非精确优化器的固有性质。