A conditional bound on sphere tangencies in all 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 $n\ge 3$. 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 $O_{\epsilon }(B^{1/n-\epsilon }{N}^{2-1/n+\epsilon })$.

10 pages, 2 figures