Improved Distribution Estimation in $\ell_\infty$

We present improved bounds for estimating discrete probability distributions under the $\ell_{\infty}$ norm. These include minimax bounds in expectation and high-probability tail bounds. We resolve some of the open questions posed in Kontorovich and Painsky (JMLR, 2025) --- including a fully empirical version of the tightest risk bound they presented and identifying the form of the worst-case extremal distribution. Encouraging empirical results are reported as well.