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On suprema of convolutions on discrete cubes

José Gaitan, José Madrid

TL;DR

The paper determines the exact optimal constant for the infinity norm of the k-fold convolution of functions on the discrete cube {0,1}^d, revealing a precise formula for C_{k,1} and its d-dimensional generalization. The authors couple a diagonal-case analysis with a dimension-compression argument, employing a Poisson Binomial distribution viewpoint to show that the extremal configuration occurs when all inputs are equal. This yields sharp bounds on Sidon-like sets on hypercubes and sheds light on the continuous analogue problem, with explicit equality cases and strong structural insights. The work advances convolution inequalities on hypercubes and connects them to additive combinatorics through tight extremal bounds.

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.

On suprema of convolutions on discrete cubes

TL;DR

The paper determines the exact optimal constant for the infinity norm of the k-fold convolution of functions on the discrete cube {0,1}^d, revealing a precise formula for C_{k,1} and its d-dimensional generalization. The authors couple a diagonal-case analysis with a dimension-compression argument, employing a Poisson Binomial distribution viewpoint to show that the extremal configuration occurs when all inputs are equal. This yields sharp bounds on Sidon-like sets on hypercubes and sheds light on the continuous analogue problem, with explicit equality cases and strong structural insights. The work advances convolution inequalities on hypercubes and connects them to additive combinatorics through tight extremal bounds.

Abstract

We find the optimal constant 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 . As applications, we derive bounds for Sidon sets on hypercubes, and, we also obtain bounds for the continuous analogue problem.

Paper Structure

This paper contains 10 sections, 10 theorems, 53 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Main results and convolution inequalities on $\{0,1\}^d$
  3. Proof of main results
  4. The diagonal case.
  5. From the general case to the diagonal case.
  6. Proof strategy, notation and basic properties.
  7. Reduction to intersection points
  8. Some auxiliary results
  9. Lagrange Multipliers and conclusion of the argument.
  10. Acknowledgements.

Key Result

Theorem 2.1

Let $k\in\mathbb{N}, k\geqslant 2$, then

Figures (1)

  • Figure 1:

Theorems & Definitions (17)

  • Theorem 2.1
  • Corollary 1
  • Theorem 2.2
  • Corollary 2
  • proof : Proof of Corollary \ref{['corollary for sumsets']}
  • proof : Proof of Theorem \ref{['thm: main theorem for d>1']}
  • proof : Proof of Theorem \ref{['thm: main theorem']}
  • Lemma 3.1
  • proof : Proof of Lemma \ref{['Key lemma todas iguales']}
  • Proposition 3.2
  • ...and 7 more