Mathematical Physics, Analysis and Geometry
, Volume 9, Issue 2, pp 95108
Geometry and Growth Rate of Frobenius Numbers of Additive Semigroups
 V. I. ArnoldAffiliated withSteklov Mathematical Institute
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Linear combinations x _{1} a _{1} ⋯ x _{ n } a _{ n } of n given natural numbers a _{ s } (with nonnegative integral coefficients x _{ s }) attain all the integral values, starting from some integer N(a), called the Frobenius number of vector a (provided that the integers a _{ s } have no common divisor, greater than 1). The growth rate of N(a) with the large value of σ = ta _{1} ⋯ a _{ n } 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 σ ^{2}. 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 σ ^{ p } 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 higherdimensional 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 domainsMathematics Subject Classification (2000)
11XX Title
 Geometry and Growth Rate of Frobenius Numbers of Additive Semigroups
 Journal

Mathematical Physics, Analysis and Geometry
Volume 9, Issue 2 , pp 95108
 Cover Date
 200605
 DOI
 10.1007/s1104000690103
 Print ISSN
 13850172
 Online ISSN
 15729656
 Publisher
 Kluwer Academic Publishers
 Additional Links
 Topics
 Keywords

 11XX
 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
 Authors

 V. I. Arnold ^{(1)}
 Author Affiliations

 1. Steklov Mathematical Institute, 8, Gubkina Street, 119991, Moscow, GSP1, Russia