New Conjectural Lower Bounds on the Optimal Density of Sphere Packings

Abstract

Sphere packings in high dimensions interest mathematicians and physicists and have direct applications in communications theory. Remarkably, no one has been able to provide exponential improvement on a hundred-year-old lower bound on the maximal packing density due to Minkowski in $d$-dimensional Euclidean space $\mathbb{R}^d$. The asymptotic behavior of this bound is controlled by $2^{-d}$ in high dimensions. Using an optimization procedure that we introduced earlier and a conjecture concerning the existence of disordered sphere packings in $\mathbb{R}^d$, we obtain a conjectural lower bound on the density whose asymptotic behavior is controlled by $2^{-0.77865\ldots \,d}$, thus providing the putative exponential improvement of Minkowski's bound. The conjecture states that a hard-core nonnegative tempered distribution is a pair correlation function of a translationally invariant disordered sphere packing in $\mathbb{R}^d$ for asymptotically large $d$ if and only if the Fourier transform of the autocovariance function is nonnegative. The conjecture is supported by two explicit analytically characterized disordered packings, numerical packing constructions in low dimensions, known necessary conditions that have relevance only in very low dimensions, and the fact that we can recover the forms of known rigorous lower bounds. A byproduct of our approach is an asymptotic conjectural lower bound on the average kissing number whose behavior is controlled by $2^{0.22134\ldots \,d}$, which is to be compared to the best known asymptotic lower bound on the individual kissing number of $2^{0.2075\ldots \,d}$. Interestingly, our optimization procedure is precisely the dual of a primal linear program devised by Cohn and Elkies to obtain upper bounds on the density, and hence has implications for linear programming bounds. This connection proves that our density estimate can never exceed the Cohn--Elkies upper bound, regardless of the validity of our conjecture.