论文标题
通过翻译的farey序列缩小目标hHosperical等分
Shrinking target horospherical equidistribution via translated Farey sequences
论文作者
论文摘要
对于$ \ operatorName {sl}（d，\ mathbb {z}）\ backslash \ propatatorName {sl}（d，d，d，\ mathbb {r}）$，$ d \ geq 2 $，我们显示任何有界的子集（与horosse soree a horosse formage a horossefe），对于$ \ mathbb {z}）\ backslash \ permatatOrname {s sl}（d，d，d，d，d，d，d，d，d，d，s sl}（d，d \ geq 2 $）目标收缩到尖尖。这种类型的等均分布正在缩小目标霍斯术等分（STHE），我们显示了几种类型的缩小靶标的STHE。我们的STHE结果将已知的结果延长了$ d = 2 $和$ \ MATHCAL {l} \ backslash \ pereratatorName {psl}（2，\ Mathbb {r}）$，其中$ \ MATHCAL {l} $是任何Cofinite Fuchsian Group，至少具有一个Cusp。 证明我们的STHE结果所需的两个关键工具是重新归一化的技术和Marklof在杰出部分对Farey序列的等分分配的结果。 For our STHE results for translated horospheres, we introduce translated Farey sequences, develop some of their geometric and dynamical properties, generalize Marklof's result by proving the equidistribution of translated Farey sequences for the same distinguished sections, and use this equidistribution of translated Farey sequences along with the renormalization technique to prove our STHE results for translated horospheres.
For a certain diagonal flow on $\operatorname{SL}(d, \mathbb{Z}) \backslash \operatorname{SL}(d, \mathbb{R})$ where $d \geq 2$, we show that any bounded subset (with measure zero boundary) of the horosphere or a translated horosphere equidistributes, under a suitable normalization, on a target shrinking into the cusp. This type of equidistribution is shrinking target horospherical equidistribution (STHE), and we show STHE for several types of shrinking targets. Our STHE results extend known results for $d=2$ and $\mathcal{L} \backslash \operatorname{PSL}(2, \mathbb{R})$ where $\mathcal{L}$ is any cofinite Fuchsian group with at least one cusp. The two key tools needed to prove our STHE results for the horosphere are a renormalization technique and Marklof's result on the equidistribution of the Farey sequence on distinguished sections. For our STHE results for translated horospheres, we introduce translated Farey sequences, develop some of their geometric and dynamical properties, generalize Marklof's result by proving the equidistribution of translated Farey sequences for the same distinguished sections, and use this equidistribution of translated Farey sequences along with the renormalization technique to prove our STHE results for translated horospheres.
