Fine-Scale Statistics for the Multidimensional Farey Sequence

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. Strombergsson and the author [Ann. Math. 172:1949–2033, 2010].

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