There exist lattices with exponentially large kissing numbers

Geometry and Topology, Number theory 1 Minute

You need to know: Set {\mathbb Z} of integers, Euclidean space {\mathbb R}^n, basis for {\mathbb R}^n, norm ||x||=\sqrt{\sum_{i=1}^n x_i^2} of x=(x_1, \dots, x_n) \in {\mathbb R}^n.

Background: A lattice L in {\mathbb R}^n is a set of the form L =\left\{\left.\sum\limits_{i=1}^n a_i v_i\,\right\vert\, a_i \in {\mathbb Z}\right\}, where v_1, \dots, v_n is a basis for {\mathbb R}^n. Let \lambda_1(L) be the length of the shortest non-zero vector in L. The kissing number \tau(L) of L is the number of vectors of length \lambda_1(L) in L. The lattice kissing number \tau_n^l in dimension n is the maximum value of \tau(L) over all lattices L in {\mathbb R}^n.

The Theorem: On 3rd February 2018, Serge Vlăduţ submitted to arxiv a paper in which he proved the existence of constant c>0 such that \tau_n^l \geq e^{cn} for any n\geq 1.

Short context: The kissing number in {\mathbb R}^n is the highest number of equal nonoverlapping spheres in {\mathbb R}^n that can touch another sphere of the same size. Determining the kissing number in various dimensions is an active area of research, see here and here. The lattice kissing number corresponds to the case when spheres form a “regular pattern”. It was a long-standing open problem whether the lattice kissing number grows exponentially with dimension. The Theorem resolves this problem affirmatively.

Links: Free arxiv version of the original paper is here, journal version is here.

Go to the list of all theorems

Unknown's avatar

Published by Bogdan Grechuk

Published

Leave a comment