Equidistribution of Farey sequences on horospheres in covers of SL(n+1,Z)\SL(n+1,R) and applications

TL;DR

This paper proves the distribution of Farey sequences on horospheres in covers of SL(n+1,Z)ackslash SL(n+1,R), extending previous statistical and Diophantine approximation results to new subsets of primitive lattice points.

Contribution

It extends the equidistribution results of Farey sequences to specific subsets in covers of SL(n+1,Z)ackslash SL(n+1,R), linking horosphere dynamics with number theory.

Findings

Established limiting distribution of Farey subsets on horospheres

Extended Marklof's statistical results to new Farey subsets

Proved Frobenius number distribution for primitive lattice point sets

Abstract

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 of the Farey sequence subsets, we extend the statistical results by Marklof regarding the full multidimensional Farey sequences, and solutions by Athreya and Ghosh to Diophantine approximation problems of Erd\H{o}s-Sz\"{u}sz-Tur\'an and Kesten. We also prove that Marklof's result on the asymptotic distribution of Frobenius numbers holds for sets of primitive lattice points of the form $\mathbf{A}$.