Abstract
Given a Boolean function f : { -1, l}(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 [24] 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)) <= 2log(parallel to(f) over cap parallel to(1,epsilon) 1(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 DNFs (for constant k). (iii) 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.