Let d be a positive integer and U ⊂ ℤ^d finite. We study

$$\beta (U): = \mathop {\inf }\limits_{\mathop {A,B \ne \phi }\limits_{{\rm{finite}}} } {{\left| {A + B + U} \right|} \over {{{\left| A \right|}^{1/2}}{{\left| B \right|}^{1/2}}}},$$

and other related quantities. We employ tensorization, which is not available for the doubling constant, ∣U + U∣/∣U∣. For instance, we show

$$\beta (U) = \left| U \right|,$$

whenever U is a subset of {0,1}^d. Our methods parallel those used for the Prékopa—Leindler inequality, an integral variant of the Brunn—Minkowski inequality.

中文翻译：

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Sumsets 基数的分析方法 Dávid Matolcsi, Imre Z. Ruzsa, George Shakan, Dmitrii Zhelezov

令d为正整数且U ⊂ ℤ ^d有限。我们学习

$$\beta (U): = \mathop {\inf }\limits_{\mathop {A,B \ne \phi }\limits_{{\rm{finite}}} } {{\left| {A + B + U} \right|} \over {{{\left| 一个 \right|}^{1/2}}{{\left| B \right|}^{1/2}}}},$$

和其他相关数量。我们采用张量化，它不适用于倍增常数 ∣ U + U ∣/∣ U ∣。例如，我们展示

$$\beta (U) = \left| 你\右|,$$

只要U是 {0,1} ^d的子集。我们的方法与用于 Prékopa-Leindler 不等式的方法相似，后者是 Brunn-Minkowski 不等式的一个积分变体。