Abstract
What is the densest Lattice sphere packing in the d-dimensional Euclidean space? In this talk we will investigate this question as dimension d goes to infinity and we will focus on the lower bounds for the best packing density, or in other words on the existence results. I will give a historical overview of the existence of dense lattice packings by H. Minkowski, E. Hlawka, C. L. Siegel, C. A. Rogers, and more recently by S. Vance and A. Venkatesh. For a long time the best asymptotic lower bounds on packing density were known for lattice packings. Recently, a new asymptotic lower bound for sphere packings in high dimensions was proven by M. Campos, M. Jensen, M. Michelin, and J. Sahasrabudhe. This new bound is proven by a combinatorial method and a dense packing guaranteed by their theorem is not necessarily a lattice packing. In the final part of the lecture I will present a recent work done in collaboration with V. Serban and N. Gargava, and Ilaria Viglino on the moments of the number of lattice points in a bounded set for random lattices constructed from a number field.
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Talk
The densest lattice packings are known in dimension .
In this talk we are interested about packings when the dimension goes to infinity. This we do with lower bounds, see
- Minkowski 1905, Hlawka 1920
- Rogers 1947
- Venkatesh 2013
- Campos, Jensen, Michelin, Sahasrabudhe 2023
The probabilistic methods are much more promising than rigid standard methods. Improved methods build on the fact that they only pick lattices from a space of lattices with special symmetries.
Concretely, we take the space of random unimodular lattices in . Then we consider a random variables and calculate the expected value. This can, surprisingly, be computed very easily.
We give some detail on how the lattice theory works. Apparently, it uses some number theory. The idea is to take lattice which have the symmetries of number fields. The details are given in Gargava, Serban, Viazovska 2023 and 2024.