On suprema of convolutions on discrete cubes

We find the optimal constant $C$ such that \begin{equation*} |f_1f_2\dotsf_{k}|{\infty}\geq C\prod{i=1}^{k}|f_i|_1 \end{equation} for functions $f_i:\{0,1{\}}^d\to \mathbb{R}$. As applications, we derive bounds for Sidon sets on hypercubes, and, we also obtain bounds for the continuous analogue problem.