math.MG — Metric Geometry

A Fourier Consequence for Lattice Kissing Numbers and Average Contacts

Contributed by Scott Kominers

We deduce a new asymptotic upper bound $K_{\mathrm{lat}}(n)\leq\left(\sqrt{2e/\pi}+o(1)\right)^n$ on the maximal lattice kissing number $K_{\mathrm{lat}}(n)$ in dimension $n$, using the explicit auxiliary functions constructed in OpenAI's "Ten Advances" preprint. The same bound holds for the average contact degree of a finite packing of congruent balls. The bound's base-$2$ exponential rate rounds to $0.3956$, matching the value extrapolated empirically by Afkhami-Jeddi, Cohn, Hartman, de Laat, and Tajdini in 2020.