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)

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>>