Bounds on generalized Frobenius numbers

Let N ≥ 2 and let 1 < a 1 < ⋯ < a N be relatively prime integers. The Frobenius number of this N -tuple is defined to be the largest positive integer that has no representation as ∑ i = 1 N a i x i where x 1 , … , x N are nonnegative integers. More generally, the s -Frobenius number is defined to be the largest positive integer that has precisely s distinct representations like this. We use techniques from the geometry of numbers to give upper and lower bounds on the s -Frobenius number for any nonnegative integer s .