论文标题
有效计数球包装
Effective counting in sphere packings
论文作者
论文摘要
鉴于Zariski密集，离散的组，$γ$的$γ$，作用于$（n + 1）$ - 尺寸双曲线空间，我们使用频谱方法来获得尖锐的渐近公式，以达到某些$γ$ -ORBITS的增长率。特别是，这使我们能够为Apollonian和（更一般）Kleinian Sphere填充计数问题获得最著名的有效错误率，也就是说，计数球的数量与受增长参数界定的半径一样。我们的方法通过两种方式扩展了Kontorovich [KON09]的方法，该方法本身就是Lax-Phillips [LP82]的轨道计数方法的扩展。首先，我们在通过技术截止和平滑操作考虑的离散子组上删除了紧凑条件。其次，我们开发了一个坐标系，该坐标系自然与球体计数问题的基础性几何形状相对应，并在这些坐标中的Casimir运算符和HAAR测量中给出结构定理。
Given a Zariski-dense, discrete group, $Γ$, of isometries acting on $(n + 1)$-dimensional hyperbolic space, we use spectral methods to obtain a sharp asymptotic formula for the growth rate of certain $Γ$-orbits. In particular, this allows us to obtain a best-known effective error rate for the Apollonian and (more generally) Kleinian sphere packing counting problems, that is, counting the number of spheres in such with radius bounded by a growing parameter. Our method extends the method of Kontorovich [Kon09], which was itself an extension of the orbit counting method of Lax-Phillips [LP82], in two ways. First, we remove a compactness condition on the discrete subgroups considered via a technical cut-off and smoothing operation. Second, we develop a coordinate system which naturally corresponds to the inversive geometry underlying the sphere counting problem, and give structure theorems on the arising Casimir operator and Haar measure in these coordinates.
