Improved Bounds on Fourier Entropy and Min-entropy

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 ⊆ [n] is sampled with probability f ˆ (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(fˆ2) ≤ C ⋅ Inf (f), where H(fˆ2) is the Shannon entropy of the Fourier distribution of f and Inf(f) is the total influence of f In this article, we present three new contributions toward the FEI conjecture: (1) Our first contribution shows that H(fˆ2) ≤ 2 ⋅ aUC⊕(f), where aUC⊕(f) is the average unambiguous parity-certificate complexity of f. This improves upon several bounds shown by Chakraborty et al. [20]. We further improve this bound for unambiguous DNFs. We also discuss how our work makes Mansour's conjecture for DNFs a natural next step toward resolution of the FEI conjecture.(2) We next consider the weaker Fourier Min-entropy-influence (FMEI) conjecture posed by O'Donnell and others [50, 53], which asks if H ∞ fˆ2) ≤ C ⋅ Inf(f), where H ∞ fˆ2) is the min-entropy of the Fourier distribution. We show H∞(fˆ2) ≤ 2⋅Cmin⊕(f), where Cmin⊕(f) is the minimum parity-certificate complexity of f. We also show that for all ε≥0, we have H∞(fˆ2)≤2 log⁡(∥fˆ∥1,ε/(1−ε)), where ∥fˆ∥1,ε 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).(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 2ω(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.