论文标题
总和的分析方法
An analytic approach to cardinalities of sumsets
论文作者
论文摘要
令$ d $为正整数，$ u \ subset \ mathbb {z}^d $有限。我们研究$$β（u）：= \ inf _ {\替换{a，b \ neq \ nequrySet \\ \ \ text {有限}}}} \ frac {| a+a+a+b+|}我们采用张力，这是不适合加倍常数的，$ | u+u |/| u | $。例如，每当$ u $是$ \ {0,1 \}^d $的子集时，我们都会显示$β（u）= | u |，$$。我们的方法与用于prékopa-leindler不平等的方法相似，这是Brunn-Minkowski不等式的整体变体。
Let $d$ be a positive integer and $U \subset \mathbb{Z}^d$ finite. We study $$β(U) : = \inf_{\substack{A , B \neq \emptyset \\ \text{finite}}} \frac{|A+B+U|}{|A|^{1/2}{|B|^{1/2}}},$$ and other related quantities. We employ tensorization, which is not available for the doubling constant, $|U+U|/|U|$. For instance, we show $$β(U) = |U|,$$ 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.
