对Minkowski球堆积密度下界推测改进的分析
Analysis of Conjectural Improvements to Minkowski's Lower Bound on the Sphere Packing Density
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中文总结 AI 辅助
本文针对Torquato与Stillinger关于高维球堆积密度下界的推测，通过超均匀对关联函数及对偶线性规划方法，证明其可实现指数级改进，为该推测提供了新证据。
中文摘要 AI 辅助
Torquato与Stillinger利用对关联函数优化框架，推测高维欧氏空间$\boldsymbol{\text{R}}^d$中球堆积最大密度的Minkowski经典下界可实现指数级改进。基于他们的可实现性推测，本文证明：在足够高维下，一类简单的超均匀对关联函数可使Minkowski下界得到多项式级改进，形式为$\boldsymbol{\text{φ}}_{\text{max}} \boldsymbol{\text{≳}} \boldsymbol{d}^\beta \boldsymbol{2}^{-\boldsymbol{d}}$，其中对每个固定的$\boldsymbol{\beta >1}$均成立。由于多项式指数可随维度增大，该族方法可连续趋近前述推测的指数级改进。本文还从Cohn–Elkies对偶线性规划上界公式独立推导出相同的指数渐近速率，证明其径向目标检验函数无法渐近排除具有Torquato–Stillinger密度标度的堆积。这些不同方法的一致性，为高维中可能存在异常致密的无序球堆积提供了新证据，并强化了Torquato–Stillinger推测下界的合理性。
英文摘要
Torquato and Stillinger used a pair-correlation-function optimization framework to conjecture an exponential improvement of Minkowski's lower bound on the maximal density of sphere packings in $\mathbb{R}^d$, with rate $2^{-(0.7786524795\ldots+o(1))d}$. Conditional on their realizability conjecture, we show that a family of hyperuniform pair correlation functions yields polynomial improvements of the form $ϕ_{\max}\gtrsim d^β2^{-d}$ for every fixed $β>1$ as $d\to\infty$. As the polynomial exponent increases with dimension, this family approaches the conjectured exponential improvement. From the Cohn--Elkies dual linear-programming upper bound, we independently derive the Torquato--Stillinger rate and show that its radial test functions cannot asymptotically exclude packings with this density scaling. For the near-contact families considered, any fixed number of shells, gaps, or radial bands can improve subexponential factors but not the leading exponential rate. To surpass this rate, we introduce an explicit hyperuniform construction based on Gauss--Radau quadrature. It contains $\lfloor(d-1)/4\rfloor$ positive delta-function shells and has rate $2^{-(0.622556248918\ldots+o(1))d}$, showing how growing radial complexity improves upon the Torquato--Stillinger rate. Finally, we prove strong duality between the unrestricted pair-correlation program and its Cohn--Elkies dual: their optimal values coincide, with no duality gap. By approximation with ordinary finite-band functions, we show that the unrestricted pair-correlation program attains the optimal Cohn--Elkies rate $2^{-(0.6044005\ldots+o(1))d}$. This result is optimal but supplies no comparable closed-form family. Together, these results support the possibility of exceptionally dense disordered sphere packings in high dimensions and strengthen the case for the Torquato--Stillinger realizability conjecture.
发表机构
- Princeton University（普林斯顿大学）
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