solidot新版网站常见问题,请点击这里查看。
消息
本文已被查看15922次
Equidistribution of Farey sequences on horospheres in covers of SL(n+1,Z)\SL(n+1,R) and applications. (arXiv:1712.03258v2 [math.DS] UPDATED)
来源于:arXiv
We establish the limiting distribution of certain subsets of Farey sequences,
i.e., sequences of primitive rational points, on expanding horospheres in
covers $\Delta\backslash\mathrm{SL}(n+1,\mathbb{R})$ of
$\mathrm{SL}(n+1,\mathbb{Z})\backslash\mathrm{SL}(n+1,\mathbb{R})$, where
$\Delta$ is a finite index subgroup of $\mathrm{SL}(n+1,\mathbb{Z})$. These
subsets can be obtained by projecting to the hyperplane
$\{(x_1,\ldots,x_{n+1})\in\mathbb{R}^{n+1}:x_{n+1}=1\}$ sets of the form
$\mathbf{A}=\bigcup_{j=1}^J\boldsymbol{a}_j\Delta$, where for all $j$,
$\boldsymbol{a}_j$ is a primitive lattice point in $\mathbb{Z}^{n+1}$. Our
method involves applying the equidistribution of expanding horospheres in
quotients of $\mathrm{SL}(n+1,\mathbb{R})$ developed by Marklof and
Str\"{o}mbergsson, and more precisely understanding how the full Farey sequence
distributes in $\Delta\backslash\mathrm{SL}(n+1,\mathbb{R})$ when embedded on
expanding horospheres as done in previous work by Marklof. For each 查看全文>>