An application of kissing numbers in sum-product estimates

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 $$\max (|\mathcal{A}+\mathcal{A}|,|\mathcal{AA}|)\gg \frac{|\mathcal{A}{|}^{4/3}}{\lceil \log |\mathcal{A}|{\rceil }^{1/3}}$$ for finite sets $\mathcal{A}$ of quaternions and of a certain family of well-conditioned matrices.