Entropy versus influence for complex functions of modulus one 2

TL;DR

Problem: whether the entropy of the Fourier spectrum $H(|Fhat|^2)$ is bounded by a constant times the influence $I(F)$ for complex-valued functions on the hypercube. Approach: construct a simple explicit unit-modulus function F on {-1,1}^n and analyze its Walsh-Fourier expansion to compute I(F) and H(|Fhat|^2). Result: I(F) = n/(n+1) and H(|Fhat|^2) > (n/(n+1)) log n, providing a counterexample to the complex-valued entropy-influence conjecture. Significance: shows that the Friedgut-Kalai conjecture does not extend to unit-modulus complex-valued functions; the example is presented in a compact, elementary form and aligns with prior observations.

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 $\frac12\log n$.