论文标题
包含大型有限集的大型副本的集合密度
The density of sets containing large similar copies of finite sets
论文作者
论文摘要
我们证明，如果$ e \ subseteq \ mathbb {r}^d $（$ d \ geq 2 $）是一个lebesgue-Measurable套件，密度大于$ \ frac {n-2} {n-1} $，则$ e $包含所有$ n $ n $ p $ p $ p $的of suppully spaleses of shore species ficess ficess fircely scaless s s scaless y salleply scaless s speaces of specess y speaces specesses of。此外，可以将“足够大”视为在所有$ p $的均匀尺寸，最小分离和直径的均匀。另一方面，我们构建了一个示例，以表明保证所有大型$ n $ - 点集所需的密度趋向于$ 1 $，价格为$ 1- o（n^{ - 1/5} \ log n）$。
We prove that if $E \subseteq \mathbb{R}^d$ ($d\geq 2$) is a Lebesgue-measurable set with density larger than $\frac{n-2}{n-1}$, then $E$ contains similar copies of every $n$-point set $P$ at all sufficiently large scales. Moreover, `sufficiently large' can be taken to be uniform over all $P$ with prescribed size, minimum separation and diameter. On the other hand, we construct an example to show that the density required to guarantee all large similar copies of $n$-point sets tends to $1$ at a rate $1- O(n^{-1/5}\log n)$.
