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

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 $\mathrm{\Psi }:[0,\frac{1}{2}]\to [0,\ln 2]$ by $\mathrm{\Psi }(t):=h\left(\frac{1+\sqrt{1-4t^2}}{2}\right)$, where $h(u):=-u\ln u-(1-u)\ln (1-u)$. We show that for every Boolean $f:(\{0,1{\}}^n,{\mu }_p^n)\to \{\pm 1\}$, $${\mathrm{E}\mathrm{n}\mathrm{t}}_p(f)\ge \sum _{k=1}^n\mathrm{\Psi }(\sqrt{q(1-q)}\cdot {\mathrm{I}\mathrm{n}\mathrm{f}}_k^{(p)}[f])\mathrm{.}$$ When $p\mathrm{\ne }\frac{1}{2}$, 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.