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：Dupré la Tour 与 Fujii 以及 Hollender、Manurangsi、Meka 和 Suksompong [ITCS'26] 近期并发的研究，受公平分配应用的驱动，将经典差异理论推广至非加性函数。由于差异理论中的许多经典技术在此设定下似乎失效，包括如 Beck-Fiala 定理 [Discrete Appl. Math '81] 的线性代数方法，能否实现可比较的非加性界仍是一个广泛开放的问题。为了更好地理解非加性差异，我们在与经典 Beck-Fiala 定理类似的稀疏设定下研究覆盖函数。我们的设定推广了加性 Beck-Fiala 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 $n$ 个物品中的每一个在所有函数中仅覆盖 $t$ 个元素，我们证明了一个构造性的差异界，该界是 $t$、颜色数 $k$ 和 $\log n$ 的多项式。