Fine-scale statistics for the multidimensional Farey sequence

Abstract

We generalize classical results on the gap distribution (and other fine-scale statistics) for the one-dimensional Farey sequence to arbitrary dimension. This is achieved by exploiting the equidistribution of horospheres in the space of lattices, and the equidistribution of Farey points in a certain subspace of the space of lattices. The argument follows closely the general approach developed by A. Str\"ombergsson and the author [Annals of Math. 172 (2010) 1949--2033].