发表机构
Brandeis University（布兰迪斯大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明多维拉格朗日谱的闭包为0与其上确界的区间，利用丢番图逼近与幺模格空间动力学的对应关系，结合horosphere膨胀平移的均匀分布完成证明。
AI 中文摘要
拉格朗日谱是数论中的经典对象，定义为当α取遍无理数时，lim inf_{q→∞} q·dist(qα, ℤ) 的值构成的集合，其结构复杂，包含离散部分、霍尔射线及介于两者之间的过渡部分。类似地可定义多维拉格朗日谱，但此前对其了解甚少。本文证明，与一维情况不同，多维拉格朗日谱的闭包等于0与其上确界之间的区间。该证明依赖于丢番图逼近与幺模格空间上动力学的对应关系，通过研究被称为动力学拉格朗日谱的拉格朗日谱的动力学对应物展开；利用该设定的高阶性质，可证明动力学拉格朗日谱等于0与其最大值之间的区间。通过应用格空间中 horosphere 的膨胀平移的均匀分布，可从完整动力学谱过渡到丢番图谱的密度。
英文摘要
The Lagrange spectrum is a classical object in number theory, defined as the set of values of $\liminf_{q\to\infty} q\, \mathrm{dist}(qα,\mathbb{Z})$ where $α$ runs through irrational numbers. It has a complicated structure, with the discrete part, Hall's ray, and a transitional part in between. One can similarly define Lagrange spectrum in the multidimensional set-up, and until now not much has been understood about it. In this paper we prove that, unlike in the one-dimensional case, the closure of the multidimensional Lagrange spectrum is equal to the interval between $0$ and its supremum. The proof relies on a correspondence between Diophantine approximation and dynamics on the space of unimodular lattices and proceeds by studying a dynamical counterpart of the Lagrange spectrum that we call dynamical Lagrange spectrum. The latter is shown to be equal to the interval between $0$ and its maximum by means of an argument utilizing the higher rank nature of the set-up. A passage from full dynamical spectrum to the density of the Diophantine spectrum is achieved by applying equidistribution of expanding translates of horospheres in the space of lattices.
Comments14 pages; the proof of Theorem 1.2 and a few misprints have been corrected
