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| Main Author: | |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2510.22478 |
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| _version_ | 1866911231681495040 |
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| author | Wang, Chenjian |
| author_facet | Wang, Chenjian |
| contents | For integers $k\geq 3,d\geq 2,$ we consider the abundance property of pinned $k$-point patterns occurring in $E\subseteq \mathbb R^d$ with positive upper density $δ(E)$. We show that for any fixed $k$-point pattern $V$, there is a set $E$ with positive upper density such that $E$ avoids all sufficiently large affine copies of $V$, with one vertex fixed at any point in $E$. However, we obtain a positive quantitative result, which states that for any fixed $E$ with positive upper density, there exists a $k$-point pattern $V,$ such that for any $x\in E$, the pinned scaling factor set
\begin{equation*}
D_x^V(E):=\{r> 0: \exists \text{ isometry } O \text{ such that }x+r\cdot O(V)\subseteq E\},
\end{equation*}
has upper density $\geq \tilde \varepsilon>0$, where constant $\tilde \varepsilon$ depends on $k,d$ and $δ(E)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_22478 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Pinned patterns and density theorems in $\mathbb R^d$ Wang, Chenjian Combinatorics Classical Analysis and ODEs For integers $k\geq 3,d\geq 2,$ we consider the abundance property of pinned $k$-point patterns occurring in $E\subseteq \mathbb R^d$ with positive upper density $δ(E)$. We show that for any fixed $k$-point pattern $V$, there is a set $E$ with positive upper density such that $E$ avoids all sufficiently large affine copies of $V$, with one vertex fixed at any point in $E$. However, we obtain a positive quantitative result, which states that for any fixed $E$ with positive upper density, there exists a $k$-point pattern $V,$ such that for any $x\in E$, the pinned scaling factor set \begin{equation*} D_x^V(E):=\{r> 0: \exists \text{ isometry } O \text{ such that }x+r\cdot O(V)\subseteq E\}, \end{equation*} has upper density $\geq \tilde \varepsilon>0$, where constant $\tilde \varepsilon$ depends on $k,d$ and $δ(E)$. |
| title | Pinned patterns and density theorems in $\mathbb R^d$ |
| topic | Combinatorics Classical Analysis and ODEs |
| url | https://arxiv.org/abs/2510.22478 |