Thickness and a gap lemma in ℝ^d

2022/04/15 by Alexia Yavicoli, Yavicoli, Alexia
Computer Science · Mathematics · #11B25 #28A78 #28A80 #Advanced Topology and Set Theory #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Mathematical Dynamics and Fractals #Metric Geometry (math.MG) #Topological and Geometric Data Analysis

paper · pdf · doi:10.48550/arxiv.2204.08428

openalex publication_date 2022/04/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We give a definition of thickness in ℝ^d that is useful even for totally disconnected sets, and prove a Gap Lemma type result. We also guarantee an interval of distances in any direction in thick compact sets, relate thick sets (for this definition of thickness) with winning sets, give a lower bound for the Hausdorff dimension of the intersection of countably many of them, a result guaranteeing the presence of large patterns, and lower bounds for the Hausdorff dimension of a set in relationship with its thickness.

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