The {\em discrepancy} of a matrix $M \in \mathbb{R}^{d \times n}$ is given by $\mathrm{DISC}(M) := \min_{\boldsymbol{x} \in \{-1,1\}^n} \|M\boldsymbol{x}\|_\infty$. An outstanding conjecture, attributed to Koml\'os, stipulates that $\mathrm{DISC}(M) = O(1)$, whenever $M$ is a Koml\'os matrix, that is, whenever every column of $M$ lies within the unit sphere. Our main result asserts that $\mathrm{DISC}(M + R/\sqrt{d}) = O(d^{-1/2})$ holds asymptotically almost surely, whenever $M \in \mathbb{R}^{d \times n}$ is Koml\'os, $R \in \mathbb{R}^{d \times n}$ is a Rademacher random matrix, $d = \omega(1)$, and $n = \tilde \omega(d^{5/4})$. We conjecture that $n = \omega(d \log d)$ suffices for the same assertion to hold. The factor $d^{-1/2}$ normalising $R$ is essentially best possible and the dependency between $n$ and $d$ is also asymptotically best possible.

翻译：矩阵 $M \in \mathbb{R}^{d \times n}$ 的{\em 差异度}定义为 $\mathrm{DISC}(M) := \min_{\boldsymbol{x} \in \{-1,1\}^n} \|M\boldsymbol{x}\|_\infty$。一个归因于Komlós的著名猜想指出，当$M$为Komlós矩阵（即$M$的每一列均位于单位球内）时，总有 $\mathrm{DISC}(M) = O(1)$。我们的主要结果断言：对于任意Komlós矩阵 $M \in \mathbb{R}^{d \times n}$，若 $R \in \mathbb{R}^{d \times n}$ 为Rademacher随机矩阵，且 $d = \omega(1)$，$n = \tilde \omega(d^{5/4})$，则 $\mathrm{DISC}(M + R/\sqrt{d}) = O(d^{-1/2})$ 以渐近概率1成立。我们推测 $n = \omega(d \log d)$ 足以保证相同结论成立。归一化因子 $d^{-1/2}$ 本质上是最优的，且 $n$ 与 $d$ 的依赖关系在渐近意义下同样达到最优。