The generalized Frobenius problem via restricted partition functions

TL;DR

This paper extends the classical Frobenius problem by analyzing the largest integers with limited representations, providing formulas, asymptotics, and quasi-polynomial expressions for these generalized values.

Contribution

It introduces formulas and asymptotic behavior for generalized Frobenius numbers and related quantities using restricted partition functions.

Findings

Formulas for large $k$ generalized Frobenius numbers

Asymptotic expansions of these values

Quasi-polynomial expressions for the restricted partition function

Abstract

Given relatively prime positive integers, $a_1,\ldots,a_n$, the Frobenius number is the largest integer with no representations of the form $a_1x_1+\cdots+a_nx_n$ with nonnegative integers $x_i$. This classical value has recently been generalized: given a nonnegative integer $k$, what is the largest integer with at most $k$ such representations? Other classical values can be generalized too: for example, how many nonnegative integers are representable in at most $k$ ways? For sufficiently large $k$, we give formulas for these values by understanding the level sets of the restricted partition function (the function $f(t)$ giving the number of representations of $t$). Furthermore, we give the full asymptotics of all of these values, as well as reprove formulas for some special cases (such as the $n=2$ case and a certain extremal family from the literature). Finally, we obtain the first two leading terms of the restricted partition function as a so-called quasi-polynomial.