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 $\mathbb{R}^d$ and any set of vectors $v_1, \cdots, v_n \in \R^d$ 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 $d \geq Ω(\log^7 n)$. 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 a ``Global Interval Tree'' data structure that simultaneously controls the deviations for all prefixes.

翻译：施泰尼茨在1913年的经典结果 \cite{Ste13} 回答了黎曼和莱维（例如 \cite{Lev05}）更早提出的问题，指出：对于 $\mathbb{R}^d$ 中的任意范数 $\|\cdot\|$ 以及满足 $\sum_{i=1}^n v_i = 0$ 的任意向量集 $v_1, \cdots, v_n \in \R^d$，存在一种排序 $π: [n] \rightarrow [n]$，使得沿该顺序的所有部分和均以 $O(d)$ 为界，即对所有 $t \in [n]$ 有 $\big\| \sum_{i=1}^t v_{π(i)} \big\| \leq O(d)$。施泰尼茨的界在一般情况下紧至常数，但对于 $\ell_2$ 范数 $\|\cdot\|_2$，人们推测最优界为 $O(\sqrt{d})$。近一个世纪后，巴纳什奇克的开创性工作 \cite{Ban12} 为 $\ell_2$ 施泰尼茨问题给出了 $O(\sqrt{d} + \sqrt{\log n})$ 的界，在 $d \geq Ω(\log n)$ 的温和假设下匹配了这一猜想。巴纳什奇克的结果是非构造性的，而此前最优的算法界由班萨尔与加格 \cite{BG17} 给出，为 $O(\sqrt{d \log n})$。本文中，我们提出一种高效算法，在略差但仍为多对数条件的 $d \geq Ω(\log^7 n)$ 下，为 $\ell_2$ 施泰尼茨问题匹配了猜想的 $O(\sqrt{d})$ 界。与先前工作相同，我们的结果可推广至更难的 $\ell_2$ 前缀差异问题。我们采用通过离散布朗运动获取所需排序的框架，并由半定规划（SDP）引导。为得到结果，我们使用了班萨尔与姜 \cite{BJ26} 提出的新技术“基于仿射谱独立性的解耦”，该技术已在贝克-菲亚拉和康洛什猜想上取得重大进展，同时结合“全局区间树”数据结构以同时控制所有前缀的偏差。