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arXiv · 2007.03493

The density of sets containing large similar copies of finite sets

Kenneth Falconer·Vjekoslav Kovač·Alexia Yavicoli
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Abstract

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 $1- O(n^{-1/5}\log n)$.

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Kenneth Falconer, Vjekoslav Kovač, Alexia Yavicoli. 2020-07-07. The density of sets containing large similar copies of finite sets. https://doi.org/10.1007/s11854-022-0231-6

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