Pinned distances and density theorems in $\mathbb R^d$
Fuente:
arXiv
Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866918133607956480 |
|---|---|
| author | Wang, Chenjian |
| author_facet | Wang, Chenjian |
| contents | We study a pinned variant of Bourgain's theorem, concerning the occurrence of affine copies of $k$-point patterns in $\mathbb{R}^d$. Focusing on the case $k=2$, which corresponds to pinned distances, we show that the classical conclusion does not extend to the pinned setting: there exist sets of positive upper density in $\mathbb{R}^d$, $d \geq 2$, such that no single pinned point determines all sufficiently large distances. However, we establish a weaker quantitative result: for every point $x$ in such a set, the pinned distance set at $x$ has (one-dimensional) positive upper density. We also construct an example demonstrating the sharpness of this bound. These findings highlight a structural distinction between global and pinned configurations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_01152 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Pinned distances and density theorems in $\mathbb R^d$ Wang, Chenjian Classical Analysis and ODEs We study a pinned variant of Bourgain's theorem, concerning the occurrence of affine copies of $k$-point patterns in $\mathbb{R}^d$. Focusing on the case $k=2$, which corresponds to pinned distances, we show that the classical conclusion does not extend to the pinned setting: there exist sets of positive upper density in $\mathbb{R}^d$, $d \geq 2$, such that no single pinned point determines all sufficiently large distances. However, we establish a weaker quantitative result: for every point $x$ in such a set, the pinned distance set at $x$ has (one-dimensional) positive upper density. We also construct an example demonstrating the sharpness of this bound. These findings highlight a structural distinction between global and pinned configurations. |
| title | Pinned distances and density theorems in $\mathbb R^d$ |
| topic | Classical Analysis and ODEs |
| url | https://arxiv.org/abs/2509.01152 |