A filtration bound for the erasure entropies of Boolean functions

For a Boolean function f and a coordinate i, the erasure entropy Eᵢ(f) is the conditional entropy of the event i∈ U given U\, where U is drawn from the spectral distribution f². The Fourier entropy-influence conjecture is equivalent to the existence of an absolute c such that every non-constant f has a coordinate with Eᵢ(f)≤ c\,Infᵢ(f). We bound Eᵢ from above by revealing the other coordinates one at a time and conditioning only on the last revealed coordinate that the spectral sample contains. The resulting quantity L^πᵢ depends on the order π, is a sum, over the steps, of a two-variable entropy of the squared increments of the Doob martingales of Eᵢf and Dᵢf, lies between Eᵢ and the binary entropy of Infᵢ, and can be minimised over all orders by a shortest-path recursion. We show that L^πᵢ=O(Infᵢ) whenever, along the order, the probability that i is pivotal both before and after the not yet revealed coordinates are resampled, times the block influence of those coordinates, is O(Infᵢ²). We also give an exact formula for Eᵢ under block composition, and show that a coordinate whose variable enters only through one AND of $k+1$ variables, pivotal with probability ρ, has L^πᵢ/Infᵢ≤2+2⁻ᵏ(2k+log₂(1/ρ)) for a suitable order. Conditioning on the last r revealed coordinates gives a decreasing family of bounds that reaches Eᵢ at $r=n-1$.