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We establish the limiting distribution of certain subsets of Farey sequences,i.e., sequences of primitive rational points, on expanding horospheres incovers $\Delta\backslash\mathrm{SL}(n+1,\mathbb{R})$ of$\mathrm{SL}(n+1,\mathbb{Z})\backslash\mathrm{SL}(n+1,\mathbb{R})$, where$\Delta$ is a finite index subgroup of $\mathrm{SL}(n+1,\mathbb{Z})$. Thesesubsets 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 $\mathbb{Z}^{n+1}$. Ourmethod involves applying the equidistribution of expanding horospheres inquotients of $\mathrm{SL}(n+1,\mathbb{R})$ developed by Marklof andStr\"{o}mbergsson, and more precisely understanding how the full Farey sequencedistributes in $\Delta\backslash\mathrm{SL}(n+1,\mathbb{R})$ when embedded onexpanding horospheres as done in previous work by Marklof. For each of theFarey sequence subsets, we extend the statistical results by Marklof regardingthe full multidimensional Farey sequences, and solutions by Athreya and Ghoshto Diophantine approximation problems of Erd\H{o}s-Sz\"{u}sz-Tur\'an andKesten. We also prove that Marklof's result on the asymptotic distribution ofFrobenius numbers holds for sets of primitive lattice points of the form$\mathbf{A}$.

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