An application of kissing numbers in sum-product estimates

TL;DR

This paper applies the concept of kissing numbers from convex geometry to improve sum-product estimates for finite sets of quaternions and certain matrices, revealing new connections between geometric and algebraic combinatorics.

Contribution

It introduces a novel application of kissing numbers to derive improved sum-product bounds for quaternions and specific matrix families.

Findings

Enhanced sum-product estimate for quaternions.

Application of geometric concepts to algebraic combinatorics.

Establishment of a new link between convex geometry and sum-product problems.

Abstract

The boundedness of the kissing numbers of convex bodies has been known to Hadwiger for long. We present an application of it to the sum-product estimate $$\max(|\mathcal{A}+\mathcal{A}|,|\mathcal{A}\mathcal{A}|)\gg\dfrac{|\mathcal{A}|^{4/3}}{\lceil\log|\mathcal{A}|\rceil^{1/3}}$$ for finite sets $\mathcal{A}$ of quaternions and of a certain family of well-conditioned matrices.