论文标题
小亚集中的大笔款项
Large sumsets from small subsets
论文作者
论文摘要
在本文中，我们开始研究添加剂组合学的新问题。基本的cauchy-davenport定理给出了一个循环p（p prime）子集的集合A+B的大小（p Prime）的大小，这只是大型结果家庭的一个例子。本文我们的目的是调查如果我们限制可能用来形成总和的一组元素的数量，会发生什么。这是我们设定要回答的问题：给定两个子集A和B，B是否具有有界尺寸的子集C，使A+C很大，甚至可能与A+B的大小相当？特别是，我们可以接近Cauchy-Davenport定理的下限吗？我们的主要结果表明，令人惊讶的是，在许多情况下，不仅可以获得通常的集合绑定的渐近版本，而且还可以获得本身。
In this paper we start to investigate a new body of questions in additive combinatorics. The fundamental Cauchy--Davenport theorem gives a lower bound on the size of a sumset A+B for subsets of the cyclic group Zp of order p (p prime), and this is just one example of a large family of results. Our aim in this paper is to investigate what happens if we restrict the number of elements of one set that we may use to form the sums. Here is the question we set out to answer: given two subsets, A and B, does B have a subset C of bounded size such that A+C is large, perhaps even comparable to the size of A+B? In particular, can we get close to the lower bound of the Cauchy--Davenport theorem? Our main results show that, rather surprisingly, in many circumstances it is possible to obtain not merely an asymptotic version of the usual sumset bound, but even the exact bound itself.
