Stolarsky-Type Inequalities in a Max-Convolution Problem

Abstract

For $m\in \mathbb{N}$, let $q_m:=\frac{\log (2m+1)}{2\log (m+1)}$. The max-convolution inequality \begin{align*} \sum_{k=0}^{2m}\left(\max_{i+j=k} x_i y_j \right)^{q_m} &\ge \left(\sum_{i=0}^{m} x_i\right)^{q_m} \left(\sum_{j=0}^{m} y_j\right)^{q_m} \end{align*}for arbitrary sequences $x_0\ge x_1\ge ...\ge x_m\ge 0,y_0\ge y_1\ge ...\ge y_m\ge 0$ implies an affirmative answer to a question of Bourgain, Dilworth, Ford, Konyagin, and Kutzarova \cite{BDFKK} on the sizes of sumsets in product sets. This inequality was proven for $m=2$ by Becker, Ivanisvili, Krachun, and Madrid \cite{BIKM} by reducing the general case to the geometric block case via a max-tie analysis. We prove the geometric block case $x=(1,t,...,t^r,0,...,0)$ and $y=(1,t,...,t^s,0,...,0)$, $t\in [0,1]$, for all $m\in \mathbb{N}$ via a comparison of Stolarsky means. Some perturbations are also verified. Finally, we prove the above inequality when one sequence has only two non-zero terms.