FKN, first proof, rewritten

About twenty years ago we wrote a paper, ”Boolean Functions whose Fourier Transform is Concentrated on the First Two Levels”, [1]. 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. Here is the main theorem of FKN [1]. Theorem 0.1. Let f : {0, 1} → {0, 1}, ‖f‖2 = p. If ∑ |S|>1 f̂ 2(S) = δ then either p < K ′δ or p > 1 −K ′δ or ‖f(x1, x2, . . . , xn) − xi‖ 2 2 ≤ Kδ for some i or ‖f(x1, x2, . . . , xn)− (1− xi)‖ 2 2 ≤ Kδ for some i. Here, K ′ and K are absolute constants. Proof: First observe that we may assume that p = 1/2: replace f by a function g : {0, 1}n+1 → {0, 1} defined by g(x1, . . . , xn, xn+1) = f(x1, . . . , xn)· 1 + χn+1 2 +(1−f(1−x1, . . . , 1−xn))· 1 − χn+1 2 and note that |g|2 = 1/2, and ∑ |S|>1 f̂ 2(S) = ∑ |S|>1 ĝ 2(S). Then prove that g is close to a dictator (or anti-dictator). If the influential coordinate for g is n+ 1 then f is necessarily close to a constant, and if the influential coordinate for g is some other i, then f too is necessarily close to a dictator or anti-dictator of the ith variable. Let f := ∑ f̂(i)χi. Note that f=1 is an odd function, i.e. f=1(S) = −f=1(Sc), and that |f|2 = ∑ f̂(i) = 1/4− δ.