On the l4 : l2 ratio of functions with restricted Fourier support

Given a subset A subset of {0,1}(n), let mu(A) be the maximal ratio between l(4) and l(2) norms of a function whose Fourier support is a subset of A.(1) 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 <= k <= n. (C) 2020 Elsevier Inc. All rights reserved.