标题： 凸性，初等方法和距离 显示英文标题

标题： Convexity, Elementary Methods, and Distances

摘要： 本文考虑了Erdős不同距离问题的一个极值版本。 对于点集$P \subset \mathbb R^d$，令$\Delta(P)$表示由$P$确定的所有欧几里得距离的集合。 我们的主要结果如下：如果$\Delta(A^d) \ll |A|^2$且$d \geq 5$，则存在$A' \subset A$使得$|A'| \geq |A|/2$，从而$|A'-A'| \ll |A| \log |A|$。 这是更一般结果的一部分，该结果指出，如果$|\Delta(A^d)|$的增长受到限制，那么$A$必须具有某种加法结构。 更具体地说，对于任何两个整数$k,n$，我们有以下信息：如果\[ | \Delta(A^{2k+3})| \leq |A|^n \]，则存在$A' \subset A$使得$|A'| \geq |A|/2$且\[ | kA'- kA'| \leq k^2|A|^{2n-3}\log|A|. \]。这些结果是Hanson所研究的二维情况结果的高维推广。 显示英文摘要

摘要： This paper considers an extremal version of the Erd\H{o}s distinct distances problem. For a point set $P \subset \mathbb R^d$, let $\Delta(P)$ denote the set of all Euclidean distances determined by $P$. Our main result is the following: if $\Delta(A^d) \ll |A|^2$ and $d \geq 5$, then there exists $A' \subset A$ with $|A'| \geq |A|/2$ such that $|A'-A'| \ll |A| \log |A|$. This is one part of a more general result, which says that, if the growth of $|\Delta(A^d)|$ is restricted, it must be the case that $A$ has some additive structure. More specifically, for any two integers $k,n$, we have the following information: if \[ | \Delta(A^{2k+3})| \leq |A|^n \] then there exists $A' \subset A$ with $|A'| \geq |A|/2$ and \[ | kA'- kA'| \leq k^2|A|^{2n-3}\log|A|. \] These results are higher dimensional analogues of a result of Hanson, who considered the two-dimensional case.