On suprema of convolutions on discrete cubes

TL;DR

This paper determines the optimal constant for the supremum of convolutions of functions on discrete hypercubes, with applications to Sidon sets and continuous analogues, advancing understanding of convolution inequalities in discrete and continuous settings.

Contribution

It establishes the exact constant for convolution supremum inequalities on discrete hypercubes and applies these results to bounds on Sidon sets and continuous problems.

Findings

Derived the optimal constant C for convolution inequalities on hypercubes.

Provided bounds for Sidon sets in hypercubes.

Extended results to continuous analogue problems.

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.