  We show that for all $A, B \subseteq \{0,1,2\}^{d}$ we have $$ |A+B|\geq
(|A||B|)^{\log(5)/(2\log(3))}. $$ We also show that for all finite $A,B \subset
\mathbb{Z}^{d}$, and any $V \subseteq\{0,1\}^{d}$ the inequality $$ |A+B+V|\geq
|A|^{1/p}|B|^{1/q}|V|^{\log_{2}(p^{1/p}q^{1/q})} $$ holds for all $p \in (1,
\infty)$, where $q=\frac{p}{p-1}$ is the conjugate exponent of $p$. All the
estimates are dimension free with the best possible exponents. We discuss
applications to various related problems.
