论文标题
大的双曲线圆圈
Large hyperbolic circles
论文作者
论文摘要
我们考虑在紧凑的双曲线表面上的公共中心圈，并且在其单位切线束上增加了半径。我们为它们的等次分配率建立了精确的渐进剂。我们的结果适用于$ \ text {sl} _2（\ mathbb {r}）$的任意元素的任何圆弧的翻译。我们的证明依赖于Ratner开创的光谱方法，随后由Burger开发，用于测量和烟节流的研究。我们进一步得出了具有紧凑的限制分布的统计极限定理，以适当地重新缩放的圆平均足够定期可观察。最后，我们遵循杜克·鲁德尼克·萨纳克（Duke-Rudnick-Sarnak）和埃斯金·麦克穆伦（Eskin-McMullen）的接近，讨论双曲线平面中经典圆问题的应用。
We consider circles of common centre and increasing radius on a compact hyperbolic surface and, more generally, on its unit tangent bundle. We establish a precise asymptotics for their rate of equidistribution. Our result holds for translates of any circle arc by arbitrary elements of $\text{SL}_2(\mathbb{R})$. Our proof relies on a spectral method pioneered by Ratner and subsequently developed by Burger in the study of geodesic and horocycle flows. We further derive statistical limit theorems, with compactly supported limiting distribution, for appropriately rescaled circle averages of sufficient regular observables. Finally, we discuss applications to the classical circle problem in the hyperbolic plane, following the approach of Duke-Rudnick-Sarnak and Eskin-McMullen.
