Geometry and Growth Rate of Frobenius Numbers of Additive Semigroups

  • V. I. Arnold
Article

Abstract

Linear combinations x1a1xnan of n given natural numbers as (with nonnegative integral coefficients xs) attain all the integral values, starting from some integer N(a), called the Frobenius number of vector a (provided that the integers as have no common divisor, greater than 1). The growth rate of N(a) with the large value of σ = ta1an depends peculiarly from the direction α of the vector a = σα. The article proves the lower bound of order \(\sigma ^{{1 + \frac{1}{{n - 1}}}}\) and the upper bound of order σ2. Both orders are reached from some directions α. The averaging of N(a) along all directions, performed for σ = 7, 19, 41 and 97, provides the values, confirming the rate σp for some p between 3/2 and 2 (for n = 3), excluding neither 3/2 nor 2, for the asymptotic behaviour at large σ. One gets check p ≈ 1, 66 for σ between 100 and 200. These unexpected results, based on some strange relations of the Frobenius numbers to the higher-dimensional continued fractions geometry, lead to many facts of this arithmetic trubulence theory, discussed in this article both as theorems and as conjectures.

Key words

geometry of numbers diophantine problems weak asymptotics continued fractions convex hulls Coxeter groups Weyl chambers Poincaré series Young diagrams averaging integer points counting in polyhedral domains 

Mathematics Subject Classification (2000)

11-XX 

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References

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    Arnold, V. I., et al.: Arnold’s Problems. Springer and Phasis, 2005, Problems 1999-8, 1999-9, and 1999-10, pp. 129–130 and 614–616.Google Scholar

Copyright information

© Springer 2006

Authors and Affiliations

  • V. I. Arnold
    • 1
  1. 1.Steklov Mathematical InstituteMoscow, GSP-1Russia

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