TL;DR
This paper studies the distribution of multidimensional Farey fractions embedded in large horospheres, extending previous work from SL(d,Z) to general lattices in SL(d,R), and identifies conditions for their uniform distribution.
Contribution
It generalizes the understanding of Farey fractions' distribution from SL(d,Z) to arbitrary lattices in SL(d,R), broadening the scope of previous results.
Findings
Established conditions for uniform distribution of Farey fractions in homogeneous spaces.
Extended asymptotic distribution results from SL(d,Z) to general lattices.
Provided a framework for analyzing Farey fractions in higher-dimensional settings.
Abstract
We embed multidimensional Farey fractions in large horospheres and explain under which conditions they become uniformly distributed in the ambient homogeneous space. This question has recently been investigated in the case of SL(d,Z) to prove the asymptotic distribution of Frobenius numbers. The present paper extends these studies to general lattices in SL(d,R).
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