2014/09/09 by Allan Greenleaf, Alex Iosevich, Greenleaf, Allan +3 · 2 citations
Computer Science · Mathematics · #28A75 #42B37 #52C10 #Artificial intelligence #Boundary (topology) #Classical Analysis and ODEs (math.CA) #Combinatorics #Combinatorics (math.CO) #Complement (music) #Computer science #Constant (computer programming) #Digital Image Processing Techniques #Dimension (graph theory) #Discrete mathematics #Equilateral triangle #FOS: Mathematics #Geometry #Hausdorff dimension #Hausdorff space #Image (mathematics) #Mathematical Dynamics and Fractals #Mathematical analysis #Mathematics #Metric Geometry (math.MG) #Point (geometry) #Point processes and geometric inequalities #Set (abstract data type) #Similarity (geometry) #math.CA #math.CO #math.MG #msc:28A75 #msc:42B37 #msc:52C10
paper · pdf · doi:10.48550/arxiv.1409.2588
18 pages, 5 figures
arxiv created 2014/09/09 · openalex publication_date 2014/09/09 · arxiv updated 2014/09/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/08
We study similarity classes of point configurations in \Rd. Given a finite collection of points, a well-known question is: How high does the Hausdorff dimension \hd(E) of a compact set E ⊂ \Bbb Rd, d ≥ 2, need to be to ensure that E contains some similar copy of this configuration? We prove results for a related problem, showing that for \hd(D) sufficiently large, E must contain many point configurations that we call k-necklaces of constant gap, generalizing equilateral triangles and rhombuses in higher dimensions. Our results extend and complement those in \citeCLP14,BIT14, where related questions were recently studied.