Pinned distances and density theorems in \mathbb R^d

2025/09/01 by C.H. Wang, Wang, Chenjian
Mathematics · #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Limits and Structures in Graph Theory #Point processes and geometric inequalities #Stochastic processes and statistical mechanics

paper · pdf · doi:10.48550/arxiv.2509.01152

openalex publication_date 2025/09/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study a pinned variant of Bourgain's theorem, concerning the occurrence of affine copies of k-point patterns in 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 ℝ^d, d ≥ 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.

Citations

• On Falconer's distance set problem in the plane

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