Pinned distances and density theorems in $\mathbb R^d$

Chenjian Wang

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arXiv · 2509.01152

Pinned distances and density theorems in $\mathbb R^d$

Chenjian Wang

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Abstract

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.

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BibTeX RIS

Chenjian Wang. 2025-09-01. Pinned distances and density theorems in $\mathbb R^d$. https://arxiv.org/abs/2509.01152

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