关于布尔平缓函数的傅里叶熵-影响猜想
On the Fourier Entropy-Influence Conjecture for Boolean Plateaued Functions
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中文总结 AI 辅助
本文通过证明一个关键不等式，解决了布尔平缓函数和部分弯曲函数的傅里叶熵-影响猜想，并确定了相应的最优常数分别为4和2。
中文摘要 AI 辅助
我们证明了布尔函数的以下不等式：$2\sum_{x\in F_2^n}f(x)wt(x)\geq wt(f)(n-deg(f))$。利用该不等式，我们建立了布尔平缓函数的傅里叶熵-影响（FEI）猜想。特别地，我们证明了平缓函数类的最优FEI常数为4。我们还证明了部分弯曲函数的FEI猜想，并表明相应的最优常数为2。最后，我们推导了布尔超立方体上p-偏置分布的若干估计。关键词：傅里叶熵，总影响，平均灵敏度，平缓函数，代数次数，里德-穆勒码，p-偏置分布。
英文摘要
We prove the following inequality for Boolean functions: $2\sum_{x\in F_2^n}f(x)wt(x)\geq wt(f)(n-deg(f))$. Using this inequality, we establish the Fourier Entropy-Influence (FEI) conjecture for Boolean plateaued functions. In particular, we show that the sharp FEI constant for the class of plateaued functions is 4. We also prove the FEI conjecture for partially bent functions and show that the corresponding sharp constant is 2. Finally, we derive several estimates for the p-biased distribution on the Boolean hypercube. Keywords: Fourier entropy, total influence, average sensitivity, plateaued function, algebraic degree, Reed-Muller code, p-biased distribution.
