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

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