Shrinking target equidistribution of horocycles in cusps
Shrinking target equidistribution of horocycles in cusps
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DOI：
10.1007/s00209-022-03118-0
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发表时间：
2021-06
影响因子：
0.8
通讯作者：
J. Tseng
中科院分区：
文献类型：
--
作者：
J. Tseng
Consider a shrinking neighborhood of a cusp of the unit tangent bundle of a noncompact hyperbolic surface of finite area, and let the neighborhood shrink into the cusp at a rate ofas. We show that a closed horocycle whose lengthgoes to infinity or even a segment of that horocycle becomes equidistributed on the shrinking neighborhood when normalized by the rateprovided thatand, for any, the segment remains larger than \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\max \left\{ T^{-1/6},\left( T/\ell \right) ^{1/2}\right\} \left( T/\ell \right) ^{-\delta }$$\end{document}. We also have an effective result for a smaller range of rates of growth ofTand. Finally, a number-theoretic identity involving the Euler totient function follows from our technique.
Consider a shrinking neighborhood of a cusp of the unit tangent bundle of a noncompact hyperbolic surface of finite area, and let the neighborhood shrink into the cusp at a rate ofas. We show that a closed horocycle whose lengthgoes to infinity or even a segment of that horocycle becomes equidistributed on the shrinking neighborhood when normalized by the rateprovided thatand, for any, the segment remains larger than \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\max \left\{ T^{-1/6},\left( T/\ell \right) ^{1/2}\right\} \left( T/\ell \right) ^{-\delta }$$\end{document}. We also have an effective result for a smaller range of rates of growth ofTand. Finally, a number-theoretic identity involving the Euler totient function follows from our technique.