arXiv 1301.00962013-01math.COmath.GR

临界积集的结构

The Structure of Critical Product Sets

Matt DeVos

备注：: 147 pages, 28 figures

摘要：

设$G$是乘法群，$A,B \subseteq G$是有限非空的，定义乘积集$AB = {ab \mid $ a \in A $ and $ b \in B $}$.组合数论中的两个基本问题是找到$|AB|$的下界，然后在$|AB|$很小的假设下确定$A$和$B$的结构性质。我们专注于当$|AB| < |A| + |B|$时的极端情况，并将任何这样的对称为$(A,B)$ \emph{critical}。在$|G|$是素数的情况下，柯西-达文波特定理断言，$|AB| \ge \min {|G|, |A| + |B| - 1}$和沃什通过对这些组中的所有临界对进行分类来改进这个结果。对于交换群，Kneser证明了Cauchy-Davenport的自然推广，证明了存在$H \le G$使得$|AB| \ge |A| + |B| - |H|$和$ABH = AB$。肯珀曼然后证明了一个结果，其特点是结构的所有关键对交换群。我们的主要结果给出了一个分类的所有临界对在任意组$G$。因此，我们得出以下推广的Kneser定理任意群：存在$H \le G$，使$|AB| \ge |A| + |B| - |H|$和，使每个$y \in AB$存在$x \in G$，使$y(x^{-1} H x) \subseteq AB$。

英文摘要：

Let $G$ be a multiplicative group, let $A,B \subseteq G$ be finite and nonempty, and define the product set $AB = {ab \mid $a \in A$ and $b \in B$}$. Two fundamental problems in combinatorial number theory are to find lower bounds on $|AB|$, and then to determine structural properties of $A$ and $B$ under the assumption that $|AB|$ is small. We focus on the extreme case when $|AB| < |A| + |B|$, and call any such pair $(A,B)$ \emph{critical}. In the case when $|G|$ is prime, the Cauchy-Davenport Theorem asserts that $|AB| \ge \min {|G|, |A| + |B| - 1}$, and Vosper refined this result by classifying all critical pairs in these groups. For abelian groups, Kneser proved a natural generalization of Cauchy-Davenport by showing that there exists $H \le G$ so that $|AB| \ge |A| + |B| - |H|$ and $ABH = AB$. Kemperman then proved a result which characterizes the structure of all critical pairs in abelian groups. Our main result gives a classification of all critical pairs in an arbitrary group $G$. As a consequence of this we derive the following generalization of Kneser's Theorem to arbitrary groups: There exists $H \le G$ so that $|AB| \ge |A| + |B| - |H|$ and so that for every $y \in AB$ there exists $x \in G$ so that $y(x^{-1} H x) \subseteq AB$.

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