A conditional bound on sphere tangencies in all dimensions

TL;DR

This paper establishes a polynomial method-based upper bound on the number of tangent pairs among N spheres in n-dimensional space, under a non-degeneracy condition limiting tangency concentration.

Contribution

It introduces a new bound on sphere tangencies in all dimensions using polynomial techniques and a non-degeneracy condition inspired by prior work.

Findings

Bound on tangent pairs is $O_{\epsilon}(B^{1/n - \epsilon} N^{2 - 1/n + \epsilon})$

Applicable for any dimension $n \\geq 3$ with a non-degeneracy condition

Extends previous 3D results to higher dimensions

Abstract

We use polynomial method techniques to bound the number of tangent pairs in a collection of $N$ spheres in $\mathbb{R}^n$ subject to a non-degeneracy condition, for any $n \geq 3$. 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 $O_{\epsilon}(B^{1/n - \epsilon} N^{2 - 1/n + \epsilon})$.