A conditional bound on sphere tangencies in all dimensions

01/16/2023

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by   Conrad Crowley, et al.

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We use polynomial method techniques to bound the number of tangent pairs in a collection of N spheres in ℝ^n 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).