Generalized Frobenius numbers: bounds and average behavior

Abstract

Let n ≥ 2 and s ≥ 1 be integers and a = (a1, . . . , an) be a relatively prime integer n-tuple. The s-Frobenius number of this ntuple, Fs(a), is defined to be the largest positive integer that cannot be represented as ∑n i=1 aixi in at least s different ways, where x1, ..., xn are non-negative integers. This natural generalization of the classical Frobenius number, F1(a), has been studied recently by a number of authors. We produce new upper and lower bounds for the s-Frobenius number by relating it to the so called s-covering radius of a certain convex body with respect to a certain lattice; this generalizes a well-known theorem of R. Kannan for the classical Frobenius number. Using these bounds, we obtain results on the average behavior of the s-Frobenius number, extending analogous recent investigations for the classical Frobenius number by a variety of authors. We also derive bounds on the s-covering radius, an interesting geometric quantity in its own right.