Given a subset A 0 1 n , let  ( A ) be the maximal ratio between 4 and 2 norms of a function whose Fourier support is a subset of A . We make some simple observations about the connections between  ( A ) and the additive properties of A on one hand, and between ( 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  ( A ) rather precisely, when A is a Hamming sphere S ( n k ) for all 0 k n .