A New Linear Programming Method in Sphere Packing

TL;DR

This paper introduces an advanced linear programming technique for sphere packing problems, extending previous methods by using sequences of auxiliary functions, and demonstrates its effectiveness in low dimensions, with potential applications in higher dimensions.

Contribution

The paper develops a new linear programming framework for sphere packing, offering greater flexibility and constructing effective auxiliary functions in low dimensions.

Findings

Effective auxiliary functions for dimensions 1, 2, 3

Extended the original linear programming approach

Potential insights for higher-dimensional sphere packing

Abstract

Inspired by the linear programming method developed by Cohn and Elkies (Ann. Math. 157(2): 689-714, 2003), we introduce a new linear programming method to solve the sphere packing problem. More concretely, we consider sequences of auxiliary functions $\{g_m\}_{m\in \mathbb{N}^{+}}$, where $g_m$ is a $m\Lambda$-periodic auxiliary function defined on $\mathbb{R}^n$, with $\Lambda$ being a given full-rank lattice in $\mathbb{R}^n$. This new method extends the original approach and offers a greater flexibility. Furthermore, using this new linear programming framework, we construct several effective auxiliary functions for dimensions $n=1,2,3$. We hope this approach provides valuable insights into solving sphere packing problems for $n=2,3$ and even higher dimensions.