The one-coordinate step in the Fourier entropy-influence conjecture

For a Boolean function f and a coordinate i let Dᵢ(f)=H(f)-12 H(fi→ 1)-12 H(fi→ -1) be the drop in Fourier entropy when xᵢ is fixed. González, MacManus and Pereyra observed that the Fourier entropy-influence conjecture follows, by induction on the dimension, if every Boolean function has a coordinate with Dᵢ(f)≤ K\,Infᵢ(f). We show that this condition is also necessary: if the conjecture holds with constant C, then every nonconstant Boolean function has a coordinate whose erasure entropy is at most C\,Infᵢ(f), and hence one with Dᵢ(f)≤ (C+2/ln 2)Infᵢ(f). The constant 2/ln 2 is the sharp bound for the Jensen–Shannon part of Dᵢ. We also give an equivalent pseudo-likelihood form of the conjecture, a bound on the influences of extremal functions, and an example showing that the condition cannot be required at every coordinate.