On ${\mathrm{\ell }}_4:{\mathrm{\ell }}_2$ ratio of functions with restricted Fourier support

Given a subset $A\subseteq \{0,1{\}}^n$, let $\mu (A)$ be the maximal ratio between ${\mathrm{\ell }}_4$ and ${\mathrm{\ell }}_2$ 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 $0\le k\le n$.