论文标题
$ \ mathrm {sl}（n）$的最佳Diophantine指数
Optimal Diophantine Exponents for $\mathrm{SL}(n)$
论文作者
论文摘要
一组对均匀空间的作用的\ emph {diophantine指数}（由Ghosh，Gorodnik和Nevo定义）量化了组上的轨道上的点，量化了近似均质空间点的复杂性。我们表明，$ \ mathrm {sl} _n（\ Mathbb {z} [1/p]）$ - 对通用上半空间$ \ mathrm {sl} _n（\ Mathbb {r}）/\ mathrm {so lie} _n（rmath）in（rmath）in（rmath）（rmath）in（rmath）（r Mathrm { $ [1,1+O（1/n）] $，在Ghosh-Gorodnik-Nevo的方法上有了显着改善，该方法使上述范围为$ [1，N-1] $。我们还表明，在\ emph {sarnak的密度假设}的假设下，指数为\ emph {optimal}，即\等于一个。尤其是结果表明，即使在基本表示的\ emph {permeDness}（Ghosh-Ghosh-Gorodnik-Nevo的工作中的关键假设）的最佳性也无法满足。该证明使用均匀空间的光谱分解，并在Eisenstein系列的本地$ l^2 $ norms上进行边界。
The \emph{Diophantine exponent} of an action of a group on a homogeneous space, as defined by Ghosh, Gorodnik, and Nevo, quantifies the complexity of approximating the points of the homogeneous space by the points on an orbit of the group. We show that the Diophantine exponent of the $\mathrm{SL}_n(\mathbb{Z}[1/p])$-action on the generalized upper half-space $\mathrm{SL}_n(\mathbb{R})/\mathrm{SO}_n(\mathbb{R})$, lies in $[1,1+O(1/n)]$, substantially improving upon Ghosh--Gorodnik--Nevo's method which gives the above range to be $[1,n-1]$. We also show that the exponent is \emph{optimal}, i.e.\ equals one, under the assumption of \emph{Sarnak's density hypothesis}. The result, in particular, shows that the optimality of Diophantine exponents can be obtained even when the \emph{temperedness} of the underlying representations, the crucial assumption in Ghosh--Gorodnik--Nevo's work, is not satisfied. The proof uses the spectral decomposition of the homogeneous space and bounds on the local $L^2$-norms of the Eisenstein series.
