The Entropy Influence Conjecture Revisited

In this paper, we prove that most of the boolean functions, f : {−1, 1}n → {−1, 1} satisfy the Fourier Entropy Influence (FEI) Conjecture due to Friedgut and Kalai (Proc. AMS’96)[2]. The conjecture says that the Entropy of a boolean function is at most a constant times the Influence of the function. The conjecture has been proven for families of functions of smaller sizes. O’donnell, Wright and Zhou (ICALP’11)[8] verified the conjecture for the family of symmetric functions, whose size is 2n+1. They are in fact able to prove the conjecture for the family of d-part symmetric functions for constant d, the size of whose is 2O(n d). Also it is known that the conjecture is true for a large fraction of polynomial sized DNFs (COLT’10)[6]. Using elementary methods we prove that a random function with high probability satisfies the conjecture with the constant as (2 + δ), for any constant δ > 0.