Nearly Tight Bounds for Proportional Group Fair Divisions and One-Sided Discrepancy

This paper studies the problem of fair division of indivisible goods among $k$ groups of $n_1,\ldots, n_k$ agents. We look at the worst downward deviation $\textit{PROP}(n_1,\ldots, n_k)$ of an agent in a group from its $1/k$-share. We improve the bounds of (Manurangsi and Meka, 2026) and show that $\textit{PROP}(n_1,\ldots, n_k) = \tildeΘ(\sqrt{n/k})$, where $n = n_1 + \ldots + n_k$ is the total number of agents. For the proof of the upper bound, we develop novel discrepancy-type tools and, in particular, a way to efficiently work with one-sided discrepancy constraints.