FKN, first proof, rewritten
Abstract
About twenty years ago we wrote a paper, "Boolean Functions whose Fourier Transform is Concentrated on the First Two Levels", FKN. In it we offered several proofs of the statement that Boolean functions f(x1,x2,…,xn), whose Fourier coefficients are concentrated on the lowest two levels are close to a constant function or to a function of the form f=xk or f=1-xk. 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.
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