The Courtade-Kumar conjecture posits that dictatorship functions maximize the mutual information between the function's output and a noisy version of its input over the Boolean hypercube. We present two significant advancements related to this conjecture. First, we resolve an open question posed by Courtade and Kumar, proving that for any Boolean function (regardless of bias), the sum of mutual information between the function's output and the individual noisy input coordinates is bounded by $1-H(α)$, where $α$ is the noise parameter of the Binary Symmetric Channel. This generalizes their previous result which was restricted to balanced Boolean functions. Second, we advance the study of the main conjecture in the high noise regime. We establish an optimal error bound of $O(λ^2)$ for the asymptotic entropy expansion, where $λ= (1-2α)^2$, improving upon the previous best-known bounds. This refined analysis leads to a sharp, linear Fourier concentration bound for highly informative functions and significantly extends the range of the noise parameter $λ$ for which the conjecture is proven to hold.

翻译：Courtade-Kumar猜想提出，在布尔超立方上，独裁函数能够最大化函数输出与其带噪输入之间的互信息。我们在此提出与该猜想相关的两项重要进展。首先，我们解决了Courtade和Kumar提出的一个开放性问题，证明了对于任意布尔函数（无论其偏置如何），函数输出与各个带噪输入坐标之间的互信息之和的上界为 $1-H(α)$，其中 $α$ 是二进制对称信道的噪声参数。该结果推广了他们先前仅限于平衡布尔函数的结论。其次，我们在高噪声区域推进了对主猜想的研究。我们为渐近熵展开建立了一个最优误差界 $O(λ^2)$，其中 $λ= (1-2α)^2$，改进了先前已知的最佳界。这一精细分析为高信息量函数导出了一个尖锐的线性傅里叶集中界，并显著扩展了该猜想已被证明成立的噪声参数 $λ$ 的范围。