半典范度下的渐近布里尔 - 诺特存在性：能量配对、切赫不等式与覆盖半径
Asymptotic Brill-Noether Existence at the Half-Canonical Degree: Energy Pairing, Cheeger Inequality and Covering Radii
AI总结：
利用数论几何技术研究图上布里尔 - 诺特存在性猜想渐近版本，在半典范度及附近证实多个图族的情况，关键工具是切赫型不等式，还应用于动力系统相关图，最后给出处理一般情况的建议。
AI中文摘要：
我们通过受数论几何启发的技术研究图上布里尔 - 诺特存在性猜想的渐近版本。在几个高度连通的图族中，我们在半典范度（及附近）证实了该猜想的渐近版本，包括偶价扩展图、固定价至少为五的几乎拉马努金图和某些随机图。关键工具是关于特定周期集相对于图的能量二次型的覆盖半径的切赫型不等式。作为应用，我们给出了与某些称为反转系统的动力系统相关图的直径下限。最后我们提出了处理该猜想渐近版本的一般建议，即超出半典范度的情况。
英文摘要：
We study asymptotic versions of the Brill-Noether existence conjecture on graphs via techniques inspired by the geometry of numbers. We confirm an asymptotic version of the conjecture at (and near) the half-canonical degree in several well-connected families of graphs. They include expander graphs of even valence, almost-Ramanujan graphs of a fixed valence at least five and certain random graphs. In particular, for any fixed $k \geq 5$, almost all simple, connected, $k$-regular graphs satisfy the Brill-Noether existence conjecture at the half-canonical degree up to a constant factor. The key tool is a Cheeger-style inequality for the covering radius of a certain periodic set with respect to the energy quadratic form associated with the graph. As an application, we lower bound the diameter of graphs associated with certain dynamical systems called reversal systems. We conclude with a suggestion to tackle the asymptotic version of the conjecture, in general, i.e. beyond half-canonical degrees.
