A new lower bound for sphere packing

We show there exists a packing of identical spheres in $\mathbb{R}^d$ with density at least $$(1-o(1))\frac{d \log d}{2^{d+1}}\, ,$$ as $d\to\infty$. This improves upon previous bounds for general $d$ by a factor of order $\log d$ and is the first asymptotically growing improvement to Rogers' bound from 1947.