Bounds on generalized Frobenius numbers

Let N ≥ 2 and let 1 < a1 < · · · < aN be relatively prime integers. The Frobenius number of this N -tuple is defined to be the largest positive integer that has no representation as PN i=1 aixi where x1, ..., xN 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.