### Integer Points in Knapsack Polytopes and $s$-Covering Radius

#### Abstract

Given a matrix $A\in \mathbb{Z}^{m\times n}$ satisfying certain regularity assumptions, we consider for a positive integer $s$ the set ${\mathcal F}_s(A)\subset \mathbb{Z}^m$ of all vectors $b\in \mathbb{Z}^m$ such that the associated *knapsack polytope*

\begin{equation*}

P(A, b)=\{ x \in \mathbb{R}^n_{\ge 0}: A x= b\}

\end{equation*}

contains at least $s$ integer points. We present lower and upper bounds on the so called diagonal $s$-Frobenius number associated to the set ${\mathcal F}_s(A)$. 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.

#### Keywords

Knapsack polytope, (diagonal) Frobenius numbers, inhomogeneous minimum, covering radius, successive minima