模块一的复杂功能的 Entropy 相对于影响 (Entropy versus influence for complex functions of modulus one)

from arxiv, Some misprints corrected. A simpler example for a function satisfying Theorem 1, due to Joe Neeman, is included. An anonymous referee wrote that theorem 2 was known to experts but not published. No hint was given for the construction. Consequently, this note will not be sent for publication (at least in the near future)

We present an example of a function $f$ from $\{-1,1\}^n$ to the unit sphere in $\mathbb{C}$ with influence bounded by $1$ and entropy of $|\hat f|^2$ larger than $\frac12\log n$. We also present an example of a function $f$ from $\{-1,1\}^n$ to $\mathbb{R}$ with $L_2$ norm $1$, $L_\infty$ norm bounded by $\sqrt{2}$, influence bounded by $1$ and entropy of $\hat f^2$ larger than $\frac12\log n$.

翻译：我们举了一个以美元为单位域的1,1美元为单位域的函数,其影响范围为$1美元,其影响范围为$2美元,其影响范围为$2美元大于$frac12/log n美元。我们还举了一个以美元为单位域的函数,其影响范围为$1,1美元至$mathbb{R},其影响范围为$2美元,其影响范围为$2美元标准美元,其影响范围为$2美元,其影响范围为$2美元,其影响范围为$2美元,其影响范围为$1美元,其影响范围为$2美元大于$frac12\log n美元。