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A new lower bound for sphere packing (2312.10026v1)

Published 15 Dec 2023 in math.MG, math.CO, and math.PR

Abstract: We show there exists a packing of identical spheres in $\mathbb{R}d$ with density at least [ (1-o(1))\frac{d \log d}{2{d+1}}\, , ] as $d\to\infty$. This improves upon previous bounds for general $d$ by a factor of order $\log d$ and is the first asymptotically growing improvement to Rogers' bound from 1947.

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Citations (3)
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Summary

  • The paper demonstrates a novel asymptotic lower bound for sphere packing density, improving past results with a (d log d)/(2^(d+1)) factor in high dimensions.
  • The authors employ a non-lattice approach using a modified Poisson process and independent set techniques in graphs to bypass traditional lattice methods.
  • The innovative method not only refines sphere packing bounds but also offers tools potentially applicable in error-correcting codes and material science.

An Analysis of "A New Lower Bound for Sphere Packing"

In the sphere packing problem, researchers aim to determine the maximum fraction of Rd\mathbb{R}^d that can be occupied by non-overlapping identical spheres. This problem is historically significant and notoriously challenging, particularly in higher dimensions due to the complexity of packing configurations. This paper presents a novel approach, achieving an improved asymptotic lower bound for the sphere packing density θ(d)\theta(d) in high dimensions. The authors demonstrate the existence of sphere packings in Rd\mathbb{R}^d with density at least (1−o(1))dlog⁡d2d+1(1-o(1))\frac{d \log d}{2^{d+1}} as d→∞d \to \infty. This represents the first substantial advance over Rogers' bound from 1947.

Major Contributions

  1. Improved Lower Bound: The research advances the lower bound for θ(d)\theta(d) by a factor of log⁡d\log d. Previous improvements to the lower bound had plateaued at better constant factors, but this paper makes an asymptotically growing improvement.
  2. Non-Lattice-Based Approach: The authors move away from traditional lattice-based methods, employing a configuration of highly disordered sphere placements using a modified Poisson process. This represents a significant conceptual shift, as most improved bounds have historically derived from lattice packings, despite speculation that disordered configurations might perform better in high dimensions.
  3. Independent Set Problem in Graphs: The method involves finding large independent sets in carefully constructed graphs, leading the authors to develop a new tool for bounding independent set sizes in graphs with controlled maximum degree and codegree conditions. This tool has potential applicability beyond the sphere packing problem, bridging a recognized gap in related combinatorial studies.

Implications and Future Directions

The results have both theoretical and practical implications. Theoretically, they provide support to the conjecture that optimal packings in high dimensions may be disordered as opposed to crystalline. Practically, these findings could influence areas such as error-correcting codes and material science, where high-density packing strategies are relevant.

Moving forward, an effort to close the exponential gap between the upper and lower bounds in high dimensions remains. The best known upper bound, provided by Kabatjanski\u\i\, and Leven\v ste\u\i n in 1978, stands significantly higher than the presented lower bound. Endeavors towards refining the independent set tool and incorporating potential non-traditional approaches may reveal further insights.

Moreover, this work extends naturally to spherical codes, which have parallel uncertainties in comprehension regarding optimal configurations. The presented methods also improve lower bounds for spherical codes, adapting efficiently to the varied geometrical structure. Notably, the case θ=π/3\theta = \pi/3, intimately linked with the kissing number problem, suggests fertile ground for continuing exploration.

Conclusion

This paper offers a substantial advancement in understanding the sphere packing problem in higher dimensions, presenting improvements that challenge long-held assumptions about optimal packing structures. Through innovative applications of combinatorial theory to geometric problems, the authors not only provide improved bounds but also contribute valuable methodologies applicable across similar problem contexts in both mathematics and physics.

Future research might explore refinements in the presented graph-theoretic technique or investigate potential improvements to constant factors that could bring these lower bounds closer to rigorous theoretical predictions.

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  1. A new lower bound for sphere packing | arXiv:2312.10026 [math.MG] (65 points, 3 comments)