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.

翻译：矩阵$M \in \mathbb{R}^{d \times n}$的{\em 偏差}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：若$M$是Komlós矩阵（即$M$的每一列均位于单位球内），则$\mathrm{DISC}(M) = O(1)$。我们的主要结果断言：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$n = \tilde \omega(d^{5/4})$时，$\mathrm{DISC}(M + R/\sqrt{d}) = O(d^{-1/2})$依渐近几乎必然成立。我们猜想当$n = \omega(d \log d)$时同一结论依然成立。归一化因子$d^{-1/2}$本质上是最优的。