Improved sphere packing lower bounds from Hurwitz lattices (Q549223)

The lattice sphere packing density \(\delta _{n, L}\) measures the greatest proportion of \(\mathbb{R}^n\) that can be filled by non-overlapping open Euclidean balls of the same radii with centers at a lattice. The author improves the asymptotic lower bound for such a density in dimensions divisible by four, proving that \[ \delta _{4m, L} \geq \zeta (4m) \frac{24 m}{2^{4m} e (1-e^{-m})}. \] In the case \(n=4m\) this improves by a constant factor the bound \(\delta _{n, L} \geq 2 \zeta (n) (n-1) 2^{-n}\) proved by \textit{K.~Ball} [Int. Math. Res. Not. 1992, 217--221 (1992; Zbl 0776.52006)], who in turn improved the previously known estimate \(1.68 n 2^{-n}\) of \textit{H.~Davenport} and \textit{C. A.~Rogers} [Duke Math. J. 14, 367--375 (1947; Zbl 0030.34602)]. The new bound is proven using \((4m)\)-dimensional Hurwitz lattice sphere packings.