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