A Lower Bound for the Fourier Entropy of Boolean Functions on the Biased Hypercube
Fan Chang
TL;DR
This work establishes a universal lower bound on the Fourier entropy ${\rm Ent}_p(f)$ of Boolean functions on the $p$-biased hypercube in terms of the squared influences: ${\rm Ent}_p(f) \ge q(1-q) \cdot \sum_{k=1}^n {\rm Inf}^{(p)}_k[f]^2$ where $q = 4p(1-p)$ and equivalently $q(1-q) = 4p(1-p)(2p-1)^2$. The authors develop a biased Fourier-analytic framework combined with random restrictions and a telescoping-entropy differentiation argument to derive the bound, linking spectral dispersion to coordinate sensitivity. As corollaries, they obtain a lower bound on the support size of the Fourier spectrum and a noise-stability-based upper bound on ${\rm S}_\varepsilon(f)$ in terms of ${\rm Ent}_p(f)$ and $p$, via constants depending on the bias. They also conjecture that the optimal constant is a function of $q$ given by the binary entropy $h(q)$, supported by dictator and parity cases, and discuss sharpness relative to existing upper bounds for the biased FEI problem.
Abstract
We study Boolean functions on the $p$-biased hypercube $(\{0,1\}^n,μ_p^n)$ through the lens of Fourier/spectral entropy, i.e., the Shannon entropy of the squared Fourier coefficients. Motivated by recent progress on upper bounds toward the Fourier-Entropy-Influence (FEI) conjecture, we prove a complementary lower bound in terms of squared influences: for every $f:(\{0,1\}^n,μ_p^n)\to \{-1,1\}$ we have $$ {\rm Ent}_p(f)\ge 4p(1-p)(2p-1)^2\cdot\sum_{k=1}^n{\rm Inf}^{(p)}_k(f)^2.$$