An Analytic Approach to Cardinalities of Sumsets Dávid Matolcsi, Imre Z. Ruzsa, George Shakan, Dmitrii Zhelezov

Let d be a positive integer and U ⊂ ℤd finite. We study $$\beta (U): = \mathop {\inf }\limits_{\mathop {A,B \ne \phi }\limits_{{\rm{finite}}} } {{\left| {A + B U} \right|} \over {{{\left| A \right|}^{1/2}}{{\left| \right|}^{1/2}}}},$$ other related quantities. employ tensorization, which is not available for the doubling constant, ∣U U∣/∣U∣. For instance, we show (U) \left| \right|,$$ whenever subset of {0,1}d. Our methods parallel those used Prékopa—Leindler inequality, an integral variant Brunn—Minkowski inequality.