FKN, first proof, rewritten

TL;DR

This paper revisits and simplifies a classic proof regarding Boolean functions with Fourier spectra concentrated on the first two levels, clarifying their structure and near-constancy.

Contribution

The authors provide a clearer, typo-free version of the original proof about the structure of Boolean functions with low-level Fourier concentration.

Findings

Boolean functions with Fourier concentration on the first two levels are close to simple functions

The rewritten proof improves clarity and correctness of the original result

The work clarifies the structure of functions with low Fourier spectrum concentration

Abstract

About twenty years ago we wrote a paper, "Boolean Functions whose Fourier Transform is Concentrated on the First Two Levels", \cite{FKN}. In it we offered several proofs of the statement that Boolean functions $f(x_1,x_2,\dots,x_n)$, whose Fourier coefficients are concentrated on the lowest two levels are close to a constant function or to a function of the form $f=x_k$ or $f=1-x_k$. Returning to the paper lately, we noticed that the presentation of the first proof is rather cumbersome, and includes several typos. In this note we rewrite that proof, as a service to the public.