Good coordinates in the Fourier entropy-influence conjecture

For a Boolean function f let Eᵢ(f)=H( 1[i∈ U] U\), where U is drawn from the spectral distribution f². The Fourier entropy-influence conjecture holds if and only if the quantity G=fminᵢ Eᵢ(f)/Infᵢ(f) is finite, and then its constant lies between G and G+2/ln 2. We study G directly. A composition identity computes Eᵢ and Infᵢ for iterated block compositions g(h,,h) exactly from the biased spectrum of g, and towers of transitive monotone gates give Gₘₒₙ≥ 4.50642846, where Gₘₒₙ is the supremum over monotone functions. On the other side we show that every width-w DNF has a coordinate with Eᵢ/Infᵢ≤ 2+log₂ e+log₂(2ʷ⁺¹-1), with no monotonicity or read restriction, and that every relevant leaf of a read-once AND/OR formula has Eᵢ/Infᵢ<π/ln 2, which is sharp. Finally we show that the coordinate of largest influence is not a valid selection rule, and that all but a vanishing fraction of the coordinates of a monotone function can have Eᵢ/Infᵢ bounded below by any constant. Nor can the ratios be averaged with weights Infᵢᵃ⁺¹ for any a≠0.