论文标题
在差异，固有的双苯胺近似和光谱间隙
On discrepancy, intrinsic Diophantine approximation, and spectral gaps
论文作者
论文摘要
在本文中，我们为代数群体在某些均匀空间上的作用的光谱差距的大小建立了界限。我们的方法基于估算合适平均操作员的操作员规范，我们开发了为这些规范建立上限和下限的技术。我们将表明，此分析问题与代数组品种上有理点的分布差异建立界限的算术问题密切相关。作为一个应用程序，我们展示了如何使用代数组的算术晶格的属性$τ$建立一个有效的限制，这些算法是$ sl（2）$的形式，使用了内在的双苯胺近似值的估计值，这些估计来自Heath-Brown-Brown-Brown-Brown-Brown对3差异性二次二次表面上的理性分析的分析。
In the present paper we establish bounds for the size of the spectral gap for actions of algebraic groups on certain homogeneous spaces. Our approach is based on estimating operator norms of suitable averaging operators, and we develop techniques for establishing both upper and lower bounds for such norms. We shall show that this analytic problem is closely related to the arithmetic problem of establishing bounds on the discrepancy of distribution for rational points on algebraic group varieties. As an application, we show how to establish an effective bound for property $τ$ of congruence subgroups of arithmetic lattices in algebraic groups which are forms of $SL(2)$, using estimates in intrinsic Diophantine approximation which follow from Heath-Brown's analysis of rational points on 3-dimensional quadratic surfaces.
