Recent concurrent work by Dupré la Tour and Fujii and by Hollender, Manurangsi, Meka, and Suksompong [ITCS'26] introduced a generalization of classical discrepancy theory to non-additive functions, motivated by applications in fair division. As many classical techniques from discrepancy theory seem to fail in this setting, including linear algebraic methods like the Beck-Fiala Theorem [Discrete Appl. Math '81], it remains widely open whether comparable non-additive bounds can be achieved. Towards a better understanding of non-additive discrepancy, we study coverage functions in a sparse setting comparable to the classical Beck-Fiala Theorem. Our setting generalizes the additive Beck-Fiala setting, rank functions of partition matroids, and edge coverage in graphs. More precisely, assuming each of the $n$ items covers only $t$ elements across all functions, we prove a constructive discrepancy bound that is polynomial in $t$, the number of colors $k$, and $\log n$.

翻译：近期，Dupré la Tour与Fujii以及Hollender、Manurangsi、Meka和Suksompong [ITCS'26] 的并行工作，受公平分配应用的驱动，将经典差异理论推广至非可加函数。由于差异理论中的许多经典技术（包括诸如贝克-菲阿拉定理 [Discrete Appl. Math '81] 的线性代数方法）在该设定下似乎失效，是否能实现可比的非可加界仍是一个广泛开放的问题。为深化对非可加性差异的理解，我们研究了稀疏设定下的覆盖函数，该设定可与经典贝克-菲阿拉定理相媲美。我们的设定推广了可加性贝克-菲阿拉设定、划分拟阵的秩函数以及图中的边覆盖。更精确地说，假设每个 $n$ 个物品在所有函数中仅覆盖 $t$ 个元素，我们证明了一个构造性差异界，该界关于 $t$、颜色数 $k$ 和 $\log n$ 呈多项式形式。