  Research on Helly-type theorems in combinatorial convex geometry has produced
volumetric versions of Helly's theorem using witness sets and quantitative
extensions of Doignon's theorem. This paper combines these philosophies and
presents quantitative Helly-type theorems for the integer lattice with
axis-parallel boxes as witness sets. Our main result shows that, while
quantitative Helly numbers for the integer lattice grow polynomially in each
fixed dimension, their variants with boxes as witness sets are uniformly
bounded. We prove several colorful and fractional variations on this theorem.
We also prove that the Helly number for $A \times A \subseteq \mathbb{R}^2$
need not be finite even when $A \subseteq \mathbb{Z}$ is a syndetic set.
