부산대학교 도서관

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 $A$. The authors observe the connections between $\mu(A)$ and the additive properties of $A$ on the one hand, and between $\mu(A)$ and the uncertainty principle for $A$ on the other. 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. Properties of the Hamming space have been studied by Y. Polyanskiy [SIAM J. Discrete Math. {\bf 33} (2019), no.~2, 731--754; MR3945798]. The results of the paper under review are relevant to the questions investigated in [op. cit.]. Moreover, in the present paper $\mu(A)$ is determined precisely when $A$ is a Hamming sphere $ S(n, k)$ for all $ 0 \leq k \leq n $.