An analytic approach to cardinalities of sumsets

Let $d$ be a positive integer and $U subset mathbb{Z}^d$ finite. We study $$beta(U) : = inf_{substack{A , B eq emptyset text{finite}}} frac{|A+B+U|}{|A|^{1/2}{|B|^{1/2}}},$$ and other related quantities. We employ tensorization, which is not available for the doubling constant, $|U+U|/|U|$. For instance, we show $$beta(U) = |U|,$$ whenever $U$ is a subset of ${0,1}^d$. Our methods parallel those used for the Prekopa-Leindler inequality, an integral variant of the Brunn-Minkowski inequality.