Generalized Frobenius numbers: Bounds and average behavior

TL;DR

This paper establishes new bounds for the s-Frobenius number by linking it to the s-covering radius, extending classical results, and analyzes its average behavior and related geometric properties.

Contribution

It generalizes Kannan's theorem to the s-Frobenius number and provides bounds on its average behavior and the s-covering radius.

Findings

New upper and lower bounds for the s-Frobenius number

Results on the average behavior of the s-Frobenius number

Bounds on the s-covering radius

Abstract

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.