An application of kissing number in sum-product estimates

Abstract

The boundedness of the kissing numbers of convex bodies has been known to Hadwiger [9] for long. We present an application of it to the sum-product estimate $$\max(\mid{\mathcal{A}+\mathcal{A}}\mid,\mid{\mathcal{A}\mathcal{A}}\mid)\gg \frac {\mid{\mathcal{A}\mid}^{4/3}}{\lceil\log\mid{\mathcal{A}\mid} \rceil^{1/3}}$$ for finite sets \({\mathcal{A}}\) of quaternions and of a certain family of well-conditioned matrices.