Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Bounds on hyperbolic sphere packings: On a conjecture by Cohn and Zhao

Maximilian Wackenhuth

Abstract

We prove sphere packing density bounds in hyperbolic space (and more generally irreducible symmetric spaces of noncompact type), which were conjectured by Cohn and Zhao and generalize Euclidean bounds by Cohn and Elkies. We work within the Bowen-Radin framework of packing density and replace the use of the Poisson summation formula in the proof of the Euclidean bound by Cohn and Elkies with an analogous formula arising from methods used in the theory of mathematical quasicrystals.

Bounds on hyperbolic sphere packings: On a conjecture by Cohn and Zhao

Abstract

We prove sphere packing density bounds in hyperbolic space (and more generally irreducible symmetric spaces of noncompact type), which were conjectured by Cohn and Zhao and generalize Euclidean bounds by Cohn and Elkies. We work within the Bowen-Radin framework of packing density and replace the use of the Poisson summation formula in the proof of the Euclidean bound by Cohn and Elkies with an analogous formula arising from methods used in the theory of mathematical quasicrystals.

Paper Structure

This paper contains 6 sections, 5 theorems, 12 equations.

Table of Contents

  1. Introduction
  2. Euclidean packings
  3. Hyperbolic packings
  4. Disclosure statement
  5. Bowen-Radin packing density framework
  6. Replacing Poisson summation

Key Result

Theorem 1

Assume that $f:\mathbb{R}^n\to \mathbb{R}$ is continuous and rapidly decayingFor precise conditions see CohnElkies2003. and satisfies Then $\triangle_{\mathbb{R}^n}(r)\leq \mathrm{vol}(B_{r})\frac{f(0)}{\widehat{f}(0)}$.

Theorems & Definitions (11)

  • Theorem 1: Cohn--Elkies
  • Theorem 2: Bowen--Radin
  • Theorem 3: Cohn--Lurie--Sarnak, Cohn--Zhao
  • Theorem 4
  • Definition 5
  • Definition 6
  • Definition 7
  • Definition 8: Björklund--Byléhn, MichaelMattiasArXiv
  • Definition 9
  • Proposition 10
  • ...and 1 more