Finding an Optimal Proximity Bound in a Very Special Scenario

Given
$A\in \mathbb{Z}^{m\times n}$
and
$\boldsymbol{b}\in \mathbb{Z}^{m}$
, we provide a sharp upper bound for the
$\ell_{\infty}$
-distance from any vertex of the polyhedron
$P(A, \boldsymbol{b})=\left\{\boldsymbol{x} \in \mathbb{R}_{\geq 0}^n: A \boldsymbol{x}=\boldsymbol{b}\right\}$
to a nearby feasible integral point under a strong assumption regarding the dimensions of the matrix
$A$
. It is hoped that this result provides motivation for conducting further research into providing more such upper bounds under certain additional assumptions.