Section: New Results

Dense sphere packing

In [27] , we consider the sphere packing problem in arbitrary dimension: what is the maximum fraction ${\Delta }_n$ of the Euclidean space ${ℝ}_n$ that can be covered by unit balls with pairwise disjoint interiors?

${\Delta }_n$ is known for only for some small values of $n$ , and when $n$ grows, we only have lower bounds. A trivial lower bound states that for every $n$, ${\Delta }_n\ge 2^{-n}$. Minkowski and Hlwaka's Theorem (1905) improves this lower bound by a factor 2: ${\Delta }_n\ge 2\times 2^{-n}$. Asymptotic improvements of this bound were obtained (from Rogers, 1947 up to Ball, 1992), all of them being of the form ${\Delta }_n\ge cn{2}^{-n}$ where $c$ is a constant.

This problem has a natural reformulation in graph theoretic terms as follows: let $G$ denote the graph whose vertices are the points of the Euclidean space and edges are pair of vertices at distance at most 2 one from the other. The independent sets of $G$ are the sphere packings: so, finding a maximum-density sphere packing is the same as finding a maximum-density independent set in this infinite graph. By using graph theoretic arguments only, Krivelevich et al. established that ${\Delta }_n\ge 0.01n{2}^{-n}$ for sufficiently large $n$.

In a recent breakthrough, Venkatesh introduced the first superlinear improvement: there are infinitely many $n$ such that ${\Delta }_n\ge cnloglogn{2}^{-n}$, where $c$ is a constant. Venkatesh's result is however non-constructive.

In this joint work with C. Bachoc and P. Moustrou, we give a constructive proof of Venkatesh's lower bound.

This study has been carried out with financial support from the French State, managed by the French National Research Agency (ANR) in the frame of the " Investments for the future " Programme IdEx Bordeaux - CPU (ANR-10-IDEX-03-02).