Sum-Product Bounds & an Inequality for the Kissing Number in Dimension 16

Abstract

We obtain an inequality for the kissing number in 16 dimensions. We do this by generalising a sum-product bound of Solymosi and Wong for quaternions to a semialgebra in dimension 16. In particular, we obtain the inequality $$k_{16}\ge \frac{{\sum }_{x\in \mathcal{R}}\left|{\mathcal{S}}_x\right|}{\left|{\bigcup }_{x\in \mathcal{R}}{\mathcal{S}}_x\right|}-1,$$ where $k_{16}$ is the 16-dimensional kissing number, and $S_x$ and $\mathcal{R}$ are sets defined below. Along the way we also obtain a sum-product bound for subsets of the octonions which are closed under taking inverses, using a similar strategy to that used for the quaternions. We use the fact that the kissing number in eight dimensions is 240 to achieve the result. Namely, we obtain the bound $$\max (|\mathcal{A}+\mathcal{A}|,|\mathcal{AA}|)\ge \frac{|\mathcal{A}{|}^{4/3}}{(1928\cdot \lceil \log |\mathcal{A}|\rceil {)}^{1/3}},$$ where $\mathcal{A}$ is a finite set of octonions such that if $x\in \mathcal{A}$, $x^{-1}\in \mathcal{A}$ also.

11 pages