Article

Mathematical Physics, Analysis and Geometry

, Volume 9, Issue 2, pp 95-108

Geometry and Growth Rate of Frobenius Numbers of Additive Semigroups

  • V. I. ArnoldAffiliated withSteklov Mathematical Institute

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Abstract

Linear combinations x 1 a 1x 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 1a 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 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