Paper 2026/2032

Tight Constants in Talagrand's Boolean Convolution Inequality

Abstract

We study the optimal dimension-free constant $C_*(a)$ in Talagrand's Boolean convolution inequality $\Pr_\mu[T_af\ge u]\le C_*(a)/(u\sqrt{\log u})$, where $T_a$ is the noise operator, $\mu$ is the uniform measure on the Boolean cube, $f\ge0$ satisfies $\mathbb{E}_\mu[f]=1$, and $u>1$. We establish explicit upper bounds for every $0<a<1$ and prove $\limsup_{a\uparrow1}C_*(a)/\sqrt{\kappa_a}<0.3577$, where $\kappa_a:=(1+a)/(1-a)$. We establish the lower bound $\liminf_{a\uparrow1}C_*(a)/\sqrt{\kappa_a}\ge0.3570$.