On the Sum of Linear Coefficients of a Boolean Valued Function

TL;DR

This paper explores a conjecture relating the sum of linear Fourier coefficients of Boolean functions to the majority function, providing alternative formalizations using discrete derivatives.

Contribution

It offers new formalizations of the Servedio-Gopalan conjecture through discrete derivative operators on Boolean functions.

Findings

Provides alternative formulations of the conjecture

Connects Fourier coefficients with discrete derivatives

Lays groundwork for future proof strategies

Abstract

Let $f:\{-1,1\}^{n}\rightarrow \{-1,1\}$ be a Boolean valued function having total degree $d$. Then a conjecture due to Servedio and Gopalan asserts that $\sum_{i=1}^{n}\widehat{f}(i)\leq \sum_{j=1}^{d}\widehat{\text{Maj}}_{d}(j)$ where $\text{Maj}_{d}$ is the majority function on $d$ bits. Here we give some alternative formalisms of this conjecture involving the discrete derivative operators on $f$.