We study the amplitude-constrained additive white Gaussian noise channel. It is well known that the capacity-achieving input distribution for this channel is discrete and supported on finitely many points. The best known bounds show that the support size of the capacity-achieving distribution is lower-bounded by a term of order $A$ and upper-bounded by a term of order $A^2$, where $A$ denotes the amplitude constraint. It was conjectured in [1] that the linear scaling is optimal. In this work, we establish a new lower bound of order $A\sqrt{\log A}$, improving the known bound and ruling out the conjectured linear scaling. To obtain this result, we quantify the fact that the capacity-achieving output distribution is close to the uniform distribution in the interior of the amplitude constraint. Next, we introduce a wrapping operation that maps the problem to a compact domain and develop a theory of best approximation of the uniform distribution by finite Gaussian mixtures. These approximation bounds are then combined with stability properties of capacity-achieving distributions to yield the final support-size lower bound.

翻译：我们研究幅度约束加性高斯白噪声信道。众所周知，该信道的容量可达输入分布是离散的，且支撑于有限个点上。目前已知的最佳界表明，容量可达分布的支撑集基数存在下界$A$量级和上界$A^2$量级，其中$A$表示幅度约束。文献[1]曾推测线性标度是最优的。本文建立了$A\sqrt{\log A}$量级的新下界，改进了已知界并排除了推测的线性标度。为得到这一结果，我们量化了以下事实：在幅度约束内部区域，容量可达输出分布接近均匀分布。随后，我们引入一种卷绕操作将问题映射至紧致域，并发展了用有限高斯混合逼近均匀分布的最佳逼近理论。最后将这些逼近界与容量可达分布的稳定性性质相结合，得到了最终的支撑集基数下界。