The asymptotic distribution of Frobenius numbers

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36 Citations (Scopus)


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.
Translated title of the contributionThe asymptotic distribution of Frobenius numbers
Original languageEnglish
Pages (from-to)179 - 207
Number of pages29
JournalInventiones Mathematicae
Issue number1
Publication statusPublished - Jul 2010

Bibliographical note

Publisher: Springer


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