On $\ell_4 : \ell_2$ ratio of functions with restricted Fourier support

Abstract

Given a subset $A \subseteq \{0,1\}^n$, let $\mu(A)$ be the maximal ratio between $\ell_4$ and $\ell_2$ norms of a function whose Fourier support is a subset of $A$. We make some simple observations about the connections between $\mu(A)$ and the additive properties of $A$ on one hand, and between $\mu(A)$ and the uncertainty principle for $A$ on the other hand. One application obtained by combining these observations with results in additive number theory is a stability result for the uncertainty principle on the discrete cube. Our more technical contribution is determining $\mu(A)$ rather precisely, when $A$ is a Hamming sphere $S(n,k)$ for all $0 \le k \le n$.