The 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 $n = \omega(d \log d)$. The factor $d^{-1/2}$ normalising $R$ is essentially best possible and the dependency between $n$ and $d$ is asymptotically best possible.

翻译：矩阵 $M \in \mathbb{R}^{d \times n}$ 的差异定义为 $\mathrm{DISC}(M) := \min_{\boldsymbol{x} \in \{-1,1\}^n} \|M\boldsymbol{x}\|_\infty$。一个著名的猜想（归功于Komlós）指出：对于任意Komlós矩阵 $M$（即 $M$ 的每一列都位于单位球内），有 $\mathrm{DISC}(M) = O(1)$。我们的主要结果证明：当 $M \in \mathbb{R}^{d \times n}$ 为Komlós矩阵，$R \in \mathbb{R}^{d \times n}$ 为Rademacher随机矩阵，$d = \omega(1)$ 且 $n = \omega(d \log d)$ 时，$\mathrm{DISC}(M + R/\sqrt{d}) = O(d^{-1/2})$ 依渐近几乎必然成立。此处归一化因子 $d^{-1/2}$ 本质上是最优的，且 $n$ 与 $d$ 的依赖关系在渐近意义上同样最优。