Entropy versus influence for complex functions of modulus one 2

Abstract

This is a simplification of a previous version of this ArXiv note. 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 $\frac{1}{2}\log n$.

The main result was probably known to experts but not published. This note will not be sent for publication