A Lower Bound for the Fourier Entropy of Boolean Functions on the Biased Hypercube

TL;DR

This paper establishes a sharp lower bound on the Fourier entropy of Boolean functions on the biased hypercube, decomposing the entropy into coordinate-wise contributions and characterizing when equality holds.

Contribution

It provides the first tight lower bound on Fourier entropy for biased Boolean functions, extending the restriction-moment framework to the biased setting.

Findings

The lower bound is tight for p ≠ 1/2 and equality holds for parity functions.

The bound decomposes entropy into coordinate-wise contributions using a new function Ψ.

The proof adapts the restriction-moment framework to the biased hypercube.

Abstract

We study Boolean functions on the $p$-biased hypercube $(\{0,1\}^n,\mu_p^n)$ through the lens of Fourier (spectral) entropy, i.e. the Shannon entropy of the squared $p$-biased Fourier coefficients. Motivated by recent restriction-based advances on upper bounds toward the Fourier-Entropy-Influence (FEI) conjecture, we prove a complementary, sharp lower bound that decomposes the entropy into coordinate-wise contributions. Let $q:=4p(1-p)$ and define $\Psi:[0,\tfrac12]\to[0,\ln 2]$ by $\Psi(t):=h\left(\frac{1+\sqrt{1-4t^2}}{2}\right)$, where $h(u):=-u\ln u-(1-u)\ln(1-u)$. We show that for every Boolean $f:(\{0,1\}^n,\mu_p^n)\to\{\pm1\}$, $$ \mathrm{Ent}_p(f) \ge \sum_{k=1}^n \Psi\left(\sqrt{q(1-q)}\cdot\mathrm{Inf}_k^{(p)}[f]\right). $$ When $p\neq \tfrac12$, this bound is tight and equality holds if and only if $f$ is a parity function. Our proof adapts the restriction-moment framework to the biased cube.