论文标题
中型子集的大笔
Large sumsets from medium-sized subsets
论文作者
论文摘要
经典的cauchy--davenport不平等给出了$ {\ Mathbb z} _p $的两个子集的大小的下限，其中$ p $是素数。本文我们的主要目的是证明这种不平等的大大加强，在形成总和时，我们只从两个子集中的每个子集中获取少数点。我们的结果之一是，存在绝对常数$ c> 0 $，这样，如果$ a $ a和$ b $是$ {\ mathbb z} _p $的子集，$ | a | a | a | = | = | b | = n \ le p/3 $，那么有子集$ a'\ a'\ subset a $ a $和$ b'$ b'\ subset a $ b'\ subset a $ | $ | a'+b'| \ ge 2n-1 $。实际上，我们证明一个人可能会喜欢任何大小：只要$ C_1 $和$ C_2 $满足$ C_1C_2 \ GE CN $，我们可以选择$ | a'| = C_1 $和$ | B'| = C_2 $。我们证明了阿伯利亚一般群体相关的结果。
The classical Cauchy--Davenport inequality gives a lower bound for the size of the sum of two subsets of ${\mathbb Z}_p$, where $p$ is a prime. Our main aim in this paper is to prove a considerable strengthening of this inequality, where we take only a small number of points from each of the two subsets when forming the sum. One of our results is that there is an absolute constant $c>0$ such that if $A$ and $B$ are subsets of ${\mathbb Z}_p$ with $|A|=|B|=n\le p/3$ then there are subsets $A'\subset A$ and $B'\subset B$ with $|A'|=|B'|\le c \sqrt{n}$ such that $|A'+B'|\ge 2n-1$. In fact, we show that one may take any sizes one likes: as long as $c_1$ and $c_2$ satisfy $c_1c_2 \ge cn$ then we may choose $|A'|=c_1$ and $|B'|=c_2$. We prove related results for general abelian groups.
