论文标题
非均匀拉伸的等均分布翻译成平滑曲线和加权dirichlet近似
Equidistribution of non-uniformly stretching translates of shrinking smooth curves and weighted Dirichlet approximation
论文作者
论文摘要
我们表明，在$ \ mathrm {diag}（e^{nt}的动作下$ \ mathbb {r}^{n+1} $中的单对晶格，翻译的任何固定尺寸的“非分级”平滑曲线或大小的尺寸$ e^{ - t} $几乎所有曲线点的收缩片段，在曲线的任何点上，都在$ the $ t the flace as as as as as as as as as as as as $ t the f the \ f ty \ f ty \ f ty \ f ty。因此，得出的是，在$ \ mathbb {r}^n $中，几乎所有非分级$ c^{2n} $ curve的加权dirichlet近似定理几乎无法改进所有点。该结果扩展了由于Shah（2009）而引起的分析曲线的相应结果，并回答了受到Davenport和Schmidt（1969）和Kleinbock and Weiss（2008）的工作启发的一些问题。
We show that under the action of $\mathrm{diag}(e^{nt},e^{-r_1(t)},\ldots,e^{-r_n(t)})\in\mathrm{SL}(n+1,\mathbb{R})$, where $r_i(t)\to\infty$, on the space of unimodular lattices in $\mathbb{R}^{n+1}$, the translates of any fixed-sized piece of a `non-degenerate' smooth curve, or a shrinking piece of size $e^{-t}$ about almost any point of the curve, get equidistributed in the space as $t\to\infty$. From this, it follows that the weighted Dirichlet approximation theorem cannot be improved for almost all points on any non-degenerate $C^{2n}$ curve in $\mathbb{R}^n$. This result extends the corresponding result for analytic curves due to Shah (2009) and answers some questions inspired by the work of Davenport and Schmidt (1969) and Kleinbock and Weiss (2008).
