Convexity, Elementary Methods, and Distances

Pub Date : 2024-02-03 DOI:10.1007/s00454-023-00625-7

Oliver Roche-Newton, Dmitrii Zhelezov

引用次数: 0

Abstract

This paper considers an extremal version of the Erdős 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 $d \ge 5$, then there exists $A' \subset A$ with $|A'| \ge |A|/2$ 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

$$\begin{aligned} | \Delta (A^{2k+3})| \le |A|^n \end{aligned}$$

then there exists $A' \subset A$ with $|A'| \ge |A|/2$ and

$$\begin{aligned} | kA'- kA'| \le k^2|A|^{2n-3}\log |A|. \end{aligned}$$

These results are higher dimensional analogues of a result of Hanson [4], who considered the two-dimensional case.

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凸性、初等方法和距离

本文研究的是厄尔多斯显著距离问题的极值版本。对于一个点集 $P \subset {\mathbb {R}}^d$, 让 $\Delta (P)$ 表示由 P 决定的所有欧氏距离的集合。我们的主要结果如下：如果 $\Delta (A^d) \ll |A|^2$ and $d \ge 5$, 那么存在 $A' \subset A$ with $|A'| \ge |A|/2$ such that $|A'-A'|ll \A| \log |A|$。这是一个更普遍的结果的一部分，它说：如果 $|\Delta (A^d)|$ 的增长受到限制，那么 A 一定具有某种加法结构。更具体地说，对于任意两个整数 k、n，我们有如下信息：如果 $$\begin{aligned}| Delta (A^{2k+3})| |le |A|^n \end{aligned}$$那么存在 $A' \subset A$ with $|A'| ge |A|/2$ 和 $$\begin{aligned}| kA'- kA'| \le k^2|A|^{2n-3}\log |A|。\end{aligned}$$这些结果是汉森[4]结果的高维类似物，汉森考虑的是二维情况。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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