Lattices with Large Kissing Number from Local Density

Let $κ_p(n)$ denote the maximum kissing number of a lattice $\mathcal{L} \subset \mathbb{R}^n$. We prove that $κ_p(n) \geq 2^{Ω(\sqrt{n})}$ for all $p\ge 1$ and that $κ_p(n) \geq Ω(α_p^n)$ for all $p > 2$, where $α_p > 1$ is a constant depending only on $p$. In particular, our former result improves