An analytic approach to cardinalities of sumsets

Let $d$ be a positive integer and $U\subset {\mathbb{Z}}^d$ finite. We study $$\beta (U):=\underset{\begin{array}{c}A,B\mathrm{\ne }\mathrm{\varnothing }\\ \text{finite}\end{array}}{\inf }\frac{\mathrm{\mid }A+B+U\mathrm{\mid }}{\mathrm{\mid }A{\mathrm{\mid }}^{1\mathrm{/}2}\mathrm{\mid }B{\mathrm{\mid }}^{1\mathrm{/}2}},$$ and other related quantities. We employ tensorization, which is not available for the doubling constant, $\mathrm{\mid }U+U\mathrm{\mid }\mathrm{/}\mathrm{\mid }U\mathrm{\mid }$. For instance, we show $$\beta (U)=\mathrm{\mid }U\mathrm{\mid },$$ whenever $U$ is a subset of $\{0,1{\}}^d$. Our methods parallel those used for the Pr'ekopa-Leindler inequality, an integral variant of the Brunn-Minkowski inequality.