An Entropic Proof of Chang's Inequality

Chang’s lemma is a useful tool in additive combinatorics and the analysis of Boolean functions. Here we give an elementary proof using entropy. The constant we obtain is tight, and we give a slight improvement in the case where the variables are highly biased. 1 The lemma For S ∈ {0, 1}, let χk : {±1} n → R denote the character χS(x) = ∏ i∈S xi . For any function f : {±1} → R, we can then define its Fourier transform f̂ : {0, 1} → R as f̂(S) = E x f(x)χS(x) = 1 2n ∑