Weak-Type Bounds for Convolution on the Boolean Hypercube

Abstract

Let $G$ be the Boolean hypercube which carries uniform measure $\lambda$, and let $T_{\mu }$ denote convolution by a finite positive measure $\mu$ on $G$. For ${\psi }_{\mu }(u)=\sup \{u\lambda (\{T_{\mu }f\ge u\}):f\ge 0,\Vert f{\Vert }_1=1\},$ we prove Talagrand's convolution conjecture (Talagrand, 1989): if ${\mu }_a=((1+a){\delta }_1/2+(1-a){\delta }_{-1}/2{)}^{\otimes n}$ and $0<a<1$, then ${\psi }_{{\mu }_a}(u)\le C_a/\sqrt{\log u}$ for every $u>1$ and $n\ge 1$, where $C_a$ depends only on $a$. The proof utilizes the reverse-heat and Boolean-bridge framework of Chen (2025) and the localized terminal-discrepancy method of Xiang and Zhang (2026). We introduce a new power coupling: each reverse edge ratio is split into two geometric powers. This choice produces a switched exponential weight which restores the exact reverse jump rate of the perturbed coordinate. The resulting endpoint comparison yields an anti-concentration profile estimate without the iterated-logarithmic factor. The proof was discovered by the Odin Automatic AI Research Agent.