On the Fourier tails of bounded functions over the discrete cube

In this paper we consider bounded real-valued functions over the discrete cube, f : −1, 1 n → [−1, 1]. Such functions arise naturally in theoretical computer science, combinatorics, and the theory of social choice. It is often interesting to understand when these functions essentially depend on few coordinates. Our main result is a dichotomy that includes a lower bound on how fast the Fourier coefficients of such functions can decay: we show that ∑_|S| > kf̂(S)^2 ⩾ exp( - O(k^2 logk)) , unless f depends essentially only on 2 O ( k ) coordinates. We also show, perhaps surprisingly, that this result is sharp up to the log k factor. p ]The same type of result has already been proven (and shown useful) for Boolean functions [Bou02, KS]. The proof of these results relies on the Booleanity of the functions, and does not generalize to all bounded functions. In this work we handle all bounded functions, at the price of a much faster tail decay. As already mentioned, this rate of decay is shown to be both roughly necessary and sufficient. p ]Our proof incorporates the use of the noise operator with a random noise rate and some extremal properties of the Chebyshev polynomials.