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.

对组合凸几何中的Helly型定理的研究使用见证集和Doignon定理的定量扩展得出了Helly定理的体积形式。本文结合了这些原理，并提出了以平行轴为见证集的整数格的定量Helly型定理。我们的主要结果表明，尽管整数格的Helly数在每个固定维度上呈多项式增长，但它们的变体（以框作为见证集）是有界的。我们证明了该定理的几个多彩的和分数阶的变化。我们还证明了$\mathrm{一种}\times \mathrm{一种}\subseteq {\mathbb{[R}}^{\mathrm{2个}}$ 即使在 $\mathrm{一种}\subseteq \mathbb{ž}$ 是一个综合集。