Abstract
Given a Boolean function f : {-1.1}(n) {-1, 1), define the Fourier distribution to be the distribution on subsets of [n], where each S subset of [n] is sampled with probability (f) over cap (S)(2). The Fourier Entropy-influence (FEI) conjecture of Friedgut and Kalai [28] seeks to relate two fundamental measures associated with the Fourier distribution: does there exist a universal constant C > 0 such that H((f) over cap (2)) = 0, we have H-infinity((f) over cap (2)) <= 2 log(parallel to(f) over cap parallel to(1,epsilon)/(1-epsilon)), where parallel to(f) over cap parallel to(1,epsilon) is the approximate spectral norm of f.
As a corollary, we verify the FMEI conjecture for the class of read-k DN Fs (for constant k). (3) Our third contribution is to better understand implications of the FEI conjecture for the structure of polynomials that 1/3-approximate a Boolean function on the Boolean cube. We pose a conjecture: no flat polynomial (whose non-zero Fourier coefficients have the same magnitude) of degree d and sparsity 2(omega(d)) can 1/3-approximate a Boolean function.
This conjecture is known to be true assuming FEI, and we prove the conjecture unconditionally (i.e., without assuming the FEI conjecture) for a class of polynomials. We discuss an intriguing connection between our conjecture and the constant for the Bohnenblust-Hille inequality, which has been extensively studied in functional analysis.