A conditional bound on sphere tangencies in all dimensions

2023/01/16 by Conrad Crowley, Crowley, Conrad, Marco Vitturi +1
Mathematics · #05D40 #14Q30 #Combinatorics (math.CO) #Computational Geometry (cs.CG) #FOS: Computer and information sciences #FOS: Mathematics #Geometric Analysis and Curvature Flows #Mathematical Dynamics and Fractals #Point processes and geometric inequalities

paper · pdf · doi:10.48550/arxiv.2301.06414

openalex publication_date 2023/01/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We use polynomial method techniques to bound the number of tangent pairs in a collection of N spheres in ℝⁿ subject to a non-degeneracy condition, for any n ≥ 3. The condition, inspired by work of Zahl for n=3, asserts that on any sphere of the collection one cannot have more than B points of tangency concentrated on any low-degree subvariety of the sphere. For collections that satisfy this condition, we show that the number of tangent pairs is O_ε(B^{1/n - ε} N^{2 - 1/n + ε}).

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