Interior of distance trees over thin Cantor sets

2025/07/10 by Yeonwook Jung, Jung, Yeonwook, Krystal Taylor +1
Mathematics · #28A75 #28A80 #Advanced Operator Algebra Research #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Geometric and Algebraic Topology #Mathematical Dynamics and Fractals

paper · pdf · doi:10.48550/arxiv.2507.07385

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

Abstract

It is known that if a compact set E in ℝ^d has Hausdorff dimension greater than (d+1)/2, then its n-chain distance set Δⁿ(E) = \(|x¹-x²|,⋯, |xⁿ- xⁿ⁺¹|)∈ ℝⁿ: xⁱ ∈ E, xⁱ≠ x^j for i≠ j \ has nonempty interior for any n∈ ℕ. In this paper, we prove that for every Cantor set K⊂ ℝ^d and for every n∈ℕ, there exists \widetildeK⊂ ℝ^d such that the pinned n-chain distance set of K× \widetildeK⊂ ℝ^2d has nonempty interior, and hence, that Δⁿ(K× \widetildeK) has nonempty interior. Our results do not depend on the Newhouse gap lemma but rather on the containment lemma recently introduced by Jung and Lai. Our results generalize three-fold: to arbitrary finite trees, to higher dimensions, and to maps that have non-vanishing partials. As an application, we provide a class of examples of Cantor sets E⊂ ℝ^2d so that for any s≥ d, dim_{\rm H}(E)= s and Δ_xⁿ(E)^∘≠ \varnothing for some x∈ E.

Citations

• Triangles in the Plane and arithmetic progressions in thick compact subsets of ℝ^d
• Nonempty interior of pinned distance and tree sets
• Interior of certain sums and continuous images of very thin Cantor sets
• A survey on Newhouse thickness, fractal intersections and patterns
• A pinned Mattila-Sjölin type theorem for product sets
• Finite Point configurations in Products of Thick Cantor sets and a Robust Nonlinear Newhouse Gap Lemma
• Finite Point Configurations and the Regular Value Theorem in a Fractal setting
• Patterns in thick compact sets
• On Falconer's distance set problem in the plane
• Interior of sums of planar sets and curves
• On the Mattila-Sjolin theorem for distance sets

Related

• Nonempty interior of pinned distance and tree sets
• Finite Point configurations in Products of Thick Cantor sets and a Robust Nonlinear Newhouse Gap Lemma
• Finite trees inside thin subsets of \Bbb R^d
• Improvement on 2-chains inside thin subsets of Euclidean spaces
• Distances in sparse sets of large Hausdorff dimension