The Fourier Entropy-Influence (FEI) conjecture states that the Fourier entropy of Boolean functions is uniformly bounded by their total influence. It has been verified for canonical examples such as disjoint tribes and for some classes of Boolean functions such as symmetric functions and read-$k$ decision trees (with a constant that depends linearly on $k$). In this note we present new classes of Boolean functions that verify the FEI conjecture. The key element is an inequality controlling the difference between the entropy of a function $f$ and the average of the entropies of $f^{\pm}$, the sub-functions obtained by setting $x_m=\pm1$ for some $m$, by the $m$-influence of $f$. If this key inequality were to hold for Boolean functions, then the full FEI conjecture would follow by induction. We introduce the notion of a stopping binary tree and observe that functions that satisfy the key inequality at the branching nodes of the tree and the FEI conjecture at the stopping nodes will satisfy the FEI conjecture. We identify some classes of functions that fit this framework: the $δ$-tribes functions, the monotone Boolean functions with the tribe separation property, and the Boolean functions with the semi-separation property, and, and, along the way, demonstrate some results that we hope the experts in this fascinating field might find useful.

翻译：傅里叶熵-影响（FEI）猜想指出，布尔函数的傅里叶熵一致地受其总影响的约束。该猜想已针对规范示例（如不相交部落）及某些布尔函数类（如对称函数和读-$k$决策树，其中常数随$k$线性变化）得到验证。本文提出了验证FEI猜想的若干新布尔函数类。关键要素是一个不等式，该不等式控制函数$f$的熵与设定$x_m=\pm1$后得到的子函数$f^{\pm}$熵的平均值之差，该差值受$f$的$m$-影响约束。若此关键不等式对布尔函数成立，则完整的FEI猜想可通过归纳法得到。我们引入停止二叉树的概念，并观察到：在树的分支节点处满足关键不等式、在停止节点处满足FEI猜想的函数将满足FEI猜想。我们识别出符合此框架的若干函数类：$\delta$-部落函数、具有部落分离性质的单调布尔函数、具有半分离性质的布尔函数，并在此过程中展示了一些结果，希望此迷人领域的专家能发现其价值。