Near-Optimal Constructive Bounds for $\ell_2$ Prefix Discrepancy and Steinitz Problems via Affine Spectral Independence

Abstract: A classical result of Steinitz from 1913 \cite{Ste13}, answering an earlier question of Riemann and Lévy (e.g., \cite{Lev05}), states that for any norm $|\cdot|$ in $\mathbb{R}<sup>d$ and any set of vectors $v_1, \cdots, v_n \in \R<sup>d$ satisfying $\sum_{i=1}ⁿ v_i = 0$, there exists an ordering $π: [n] \rightarrow [n]$ such that every partial sum along this order is bounded by $O(d)$, i.e., $$\big| \sum_{i=1}^t v_{π(i)} \big| \leq O(d)$$ for all $t \in [n]$. Steinitz's bound is tight up to constants in general, but for the $\ell_2$ norm $|\cdot|_2$, it has been conjectured that the best bound is $O(\sqrt{d})$. Almost a century later, a breakthrough work of Banaszczyk \cite{Ban12} gave a bound of $O(\sqrt{d} + \sqrt{\log n})$ for the $\ell_2$ Steinitz problem, matching the conjecture under the mild assumption that $d \geq Ω(\log n)$. Banaszczyk's result is non-constructive, and the previous best algorithmic bound was $O(\sqrt{d \log n})$, due to Bansal and Garg \cite{BG17}. In this work, we give an efficient algorithm that matches the conjectured $O(\sqrt{d})$ bound for the $\ell_2$ Steinitz problem under the slightly worse, yet still polylogarithmic, condition of $d \geq Ω(\log⁷ n)$. As in prior work, our result extends to the harder problem of $\ell_2$ prefix discrepancy. We employ the framework of obtaining the desired ordering via a discrete Brownian motion, guided by a semidefinite program (SDP). To obtain our results, we use the new technique of Decoupling via Affine Spectral Independence'', proposed by Bansal and Jiang \cite{BJ26} to achieve substantial progress on the Beck-Fiala and Komlós conjectures, together with aGlobal Interval Tree'' data structure that simultaneously controls the deviations for all prefixes.