  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 $0 \le k \le n$.
