Integer Points in Knapsack Polytopes and s-Covering Radius

Given a matrix A ∈ Zm×n satisfying certain regularity assumptions, we consider for a positive integer s the set Fs(A) ⊂ Zm of all vectors b ∈ Zm such that the associated knapsack polytope P (A, b) = {x ∈ R>0 : Ax = b} contains at least s integer points. We present lower and upper bounds on the so called diagonal s-Frobenius number associated to the set Fs(A). In the case m = 1 we prove an optimal lower bound for the s-Frobenius number, which is the largest integer b such that P (A, b) contains less than s integer points.