On suprema of convolutions on discrete cubes

Abstract: We find the optimal constant $C$ such that \begin{equation*} |f_1*f_2*\dots*f_{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.