, Volume 9, Issue 2, pp 95-108
Date: 20 Oct 2006

Geometry and Growth Rate of Frobenius Numbers of Additive Semigroups

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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.

Partially supported by RFBR grant 05-01-00104.