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The asymptotic distribution of Frobenius numbers

Inventiones mathematicae volume 181, pages 179–207 (2010)Cite this article

Abstract

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.

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References

  1. Aliev, I.M., Gruber, P.M.: An optimal lower bound for the Frobenius problem. J. Number Theory 123, 71–79 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  2. Aliev, I.M., Henk, M.: Integer knapsacks: Average behavior of the Frobenius numbers. Math. Oper. Res. 34, 698–705 (2009)

    Article  MathSciNet  Google Scholar 

  3. Arnold, V.I.: Weak asymptotics of the numbers of solutions of Diophantine equations. Funct. Anal. Appl. 33, 292–293 (1999)

    Article  MathSciNet  Google Scholar 

  4. Arnold, V.I.: Arithmetical turbulence of selfsimilar fluctuations statistics of large Frobenius numbers of additive semigroups of integers. Mosc. Math. J. 7, 173–193 (2007)

    MATH  MathSciNet  Google Scholar 

  5. Beihoffer, D., Hendry, J., Nijenhuis, A., Wagon, S.: Faster algorithms for Frobenius numbers. Electron. J. Combin. 12, R27 (2005), 38 pp.

    MathSciNet  Google Scholar 

  6. Boothby, W.M.: An Introduction to Differentiable Manifolds and Riemannian Geometry. Academic Press, San Diego (1986)

    MATH  Google Scholar 

  7. Bourgain, J., Sinai, Ya.G.: Limit behavior of large Frobenius numbers. Russ. Math. Surv. 62, 713–725 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  8. Brauer, A., Shockley, J.E.: On a problem of Frobenius. J. Reine Angew. Math. 211, 215–220 (1962)

    MATH  MathSciNet  Google Scholar 

  9. Kannan, R.: Lattice translates of a polytope and the Frobenius problem. Combinatorica 12, 161–177 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  10. Marklof, J.: The n-point correlations between values of a linear form. Ergod. Theory Dyn. Syst. 20, 1127–1172 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  11. Marklof, J., Strömbergsson, A.: The distribution of free path lengths in the periodic Lorentz gas and related lattice point problems. Ann. Math. (to appear). arXiv:0706.4395

  12. Raghunathan, M.S.: Discrete Subgroups of Lie Groups. Springer-Verlag, New York (1972)

    MATH  Google Scholar 

  13. Ramirez Alfonsin, J.L.: The Diophantine Frobenius Problem. Oxford University Press, Oxford (2005)

    Book  MATH  Google Scholar 

  14. Rødseth, Ø.: On a linear Diophantine problem of Frobenius. J. Reine Angew. Math. 301, 171–178 (1978)

    MathSciNet  Google Scholar 

  15. Rødseth, Ø.: An upper bound for the h-range of the postage stamp problem. Acta Arith. 54, 301–306 (1990)

    MathSciNet  Google Scholar 

  16. Scarf, H.E., Shallcross, D.F.: The Frobenius problem and maximal lattice free bodies. Math. Oper. Res. 18, 511–515 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  17. Schmidt, W.M.: The distribution of sublattices of ℤm. Monatsh. Math. 125, 37–81 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  18. Selmer, E.S.: On the linear Diophantine problem of Frobenius. J. Reine Angew. Math. 293/294, 1–17 (1977)

    MathSciNet  Google Scholar 

  19. Selmer, E.S., Beyer, O.: On the linear Diophantine problem of Frobenius in three variables. J. Reine Angew. Math. 301, 161–170 (1978)

    MATH  MathSciNet  Google Scholar 

  20. Shiryaev, A.N.: Probability. Graduate Texts in Mathematics, vol. 95. Springer-Verlag, Berlin (1996)

    Google Scholar 

  21. Shur, V., Sinai, Ya.G., Ustinov, A.: Limiting distribution of Frobenius numbers for n=3. J. Number Theory 129, 2778–2789 (2009)

    Article  MathSciNet  Google Scholar 

  22. Siegel, C.L.: Lectures on the Geometry of Numbers. Springer-Verlag, Berlin (1989)

    MATH  Google Scholar 

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Authors and Affiliations

  1. School of Mathematics, University of Bristol, Bristol, BS8 1TW, UK

    Jens Marklof

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  1. Jens Marklof
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Correspondence to Jens Marklof.

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The author is supported by a Royal Society Wolfson Research Merit Award.

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Marklof, J. The asymptotic distribution of Frobenius numbers. Invent. math. 181, 179–207 (2010). https://doi.org/10.1007/s00222-010-0245-z

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  • DOI: https://doi.org/10.1007/s00222-010-0245-z

Keywords

  • Compact Subset
  • Asymptotic Distribution
  • Measure Zero
  • Lebesgue Measure Zero
  • Dynamical Interpretation