The asymptotic distribution of Frobenius numbers
By Jens Marklof. 2009
85 citations recorded by the source corpus.
The Frobenius number F(a) of an integer vector a with positive coprime coefficients is defined as the largest number that does not have a representation as a positive integer linear combination of the coefficients of a. We show that if a is taken to be random in an expanding d-dimensional domain, then F(a) has a limit distribution, which is given by the probability distribution for the covering radius of a certain simplex with respect to a (d−1)-dimensional random lattice. This result extends recent studies for d=3 by Arnold, Bourgain-Sinai and Shur-Sinai-Ustinov. The key features of our approach are (a) a novel interpretation of the Frobenius number in terms of the dynamics of a certain group action on the space of d-dimensional lattices, and (b) an equidistribution theorem for a multidimensional Farey sequence on closed horospheres.
Subfields: Geometry and Topology, Mathematical Physics, Statistics and Probability
View Paper: https://www.semanticscholar.org/paper/7be94d081b912de95a0b2d275150ec875e36e99e
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In Geometry and Topology
In Mathematical Physics
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Citations in the corpus
- A Poincar\'e section for horocycle flow on the space of lattices (2012, background): “3 can be deduced from Theorem 6 in [28], which is proven in the more general context of horospherical flows using a section that reduces to the same one in Theorem 1.”
- Fine-Scale Statistics for the Multidimensional Farey Sequence (2012, background): “The identification of (21) as an embedded submanifold, which is transversal to closed horospheres of large volume, is central to the proof of Theorem 2 in [12].”