Integer Knapsacks: Average Behavior of the Frobenius Numbers

Given a primitive integer vector a ∈ Z>0, the largest integer b such that the knapsack polytope P = {x ∈ R≥0 : 〈a,x〉 = b} contains no integer point is called the Frobenius number of a. We show that the asymptotic growth of the Frobenius number in average is significantly slower than the growth of the maximum Frobenius number. More precisely, we prove that it does not essentially exceed ||a|| 1+1/(N−1) ∞ , where || · ||∞ denotes the maximum norm.