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

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.