In this paper, we prove that the Fourier entropy of an $n$-dimensional boolean function $f$ can be upper-bounded by $O(I(f)+ \sum\limits_{k\in[n]}I_k(f)\log \frac{1}{I_k(f)})$, where $I(f)$ is its total influence and $I_k(f)$ is the influence of the $k$-th coordinate. The proof is elementary and uses iterative bounds on moments of Fourier coefficients over different levels.
A Lower Bound for the Fourier Entropy of Boolean Functions on the Biased Hypercube
School of Statistics and Data Science, Nankai University（南开大学统计与数据科学学院）；Extremal Combinatorics and Probability Group (ECOPRO), Institute for Basic Science (IBS)（基础科学研究所极组合与概率组）
Towards a Proof of the Fourier--Entropy Conjecture?
The total influence of a function is a central notion in analysis of Boolean functions, and characterizing functions that have small total influence is one of the most fundamental questions associated with it. The KKL theorem and the Friedgut junta theorem give a strong characterization of such functions whenever the bound on the total influence is $o(\log n)$ . However, both results become useless when the total influence of the function is $ω(\log n)$ . The only case in which this logarithmic barrier has been broken for an interesting class of functions was proved by Bourgain and Kalai, who …
On the Fourier Entropy Influence Conjecture for Extremal Classes
The Fourier Entropy-Influence (FEI) Conjecture of Friedgut and Kalai states that ${\bf H}[f] \leq C \cdot {\bf I}[f]$ holds for every Boolean function $f$ , where ${\bf H}[f]$ denotes the spectral entropy of $f$ , ${\bf I}[f]$ is its total influence, and $C > 0$ is a universal constant. Despite significant interest in the conjecture it has only been shown to hold for some classes of Boolean functions such as symmetric functions and read-once formulas. In this work, we prove the conjecture for extremal cases, functions with small influence and functions with high entropy. Specifically, we show …
A Multiplicative Fourier Proof of the Length-Four Index Conjecture
School of Mathematics and Statistics, Guangdong University of Foreign Studies（广东外语外贸大学数学与统计学院）；School of Mathematical Sciences, South China Normal University（华南师范大学数学科学学院）；School of Mathematics and Information Science, Guangzhou University（广州大学数学与信息科学学院）
