Nonempty interior of pinned distance and tree sets

2025/03/19 by Tainara Borges, Benjamin R. Foster, Benjamin Foster +7 · 3 citations
Computer Science · Mathematics · #Advanced Graph Theory Research #Advanced Topology and Set Theory #Computational Geometry and Mesh Generation #Dimension (graph theory) #Hausdorff distance #Hausdorff space #Point (geometry) #Set (abstract data type) #Tree (set theory) #math.CA

paper · pdf · doi:10.48550/arxiv.2503.15709

openalex publication_date 2025/03/19 · openalex created_date 2025/10/17 · openalex updated_date 2026/08/05

Abstract

For a compact set E⊂ℝ^d, d≥ 2, consider the pinned distance set Δ^y(E)=\lbrace |x-y| : x∈ E\rbrace. Peres and Schlag showed that if the Hausdorff dimension of E is bigger than (d+2)/(2) with d≥ 3, then there exists a point y∈ E such that Δ^y(E) has nonempty interior. In this paper we obtain the first non-trivial threshold for this problem in the plane, improving on the Peres--Schlag threshold when d=3, and we extend the results to trees using a novel induction argument.

Cited by

• Pinned nonempty interior and volumes of simplices
• Interior of distance trees over thin Cantor sets
• Triangles in the Plane and arithmetic progressions in thick compact subsets of ℝ^d

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