Sharp lower bound constant h(q) for spectral entropy on the biased hypercube

Establish the optimal constant h(q) = −q log q − (1−q) log(1−q) in the lower bound Ent_p(f) ≥ h(q) · ∑_{k=1}^n Inf_k^{(p)}[f]^2 for all 0 < p < 1 and all Boolean functions f: ( {0,1}^n, μ_p^n ) → {−1,1}, where q = 4p(1−p).

Background

The paper proves a general lower bound on spectral entropy in terms of squared influences with constant q(1−q), and conjectures that the optimal constant should be h(q), the binary entropy of q = 4p(1−p).

Evidence is provided via exact computations for dictatorship and parity functions, where the lower bound with h(q) is tight. The conjecture proposes a refinement of the proven lower bound, particularly improving constants in extreme-bias regimes.

References

Motivated by the exact computations below, we conjecture that the optimal constant in our lower bound depends on the bias only through the quantity $q:=4p(1-p)\in[0,1]$, and is given by the binary entropy function $h(q)=-q\log q-(1-q)\log(1-q)$. For every Boolean $f:({0,1}n,\mu_pn)\to{\pm1}$ and every $0<p<1$, ${\rm Ent}p(f) \ge h(q)\cdot \sum{k=1}n {\rm Inf}_k{(p)}[f]2."

— A Lower Bound for the Fourier Entropy of Boolean Functions on the Biased Hypercube (2511.07739 - Chang, 11 Nov 2025) in Section 4, Concluding Remarks (Conjecture [Sharp lower bound of Fourier entropy])