Given an $m\times n$ binary matrix $M$ with $|M|=p\cdot mn$ (where $|M|$ denotes the number of 1 entries), define the discrepancy of $M$ as $\mbox{disc}(M)=\displaystyle\max_{X\subset [m], Y\subset [n]}\big||M[X\times Y]|-p|X|\cdot |Y|\big|$. Using semidefinite programming and spectral techniques, we prove that if $\mbox{rank}(M)\leq r$ and $p\leq 1/2$, then $$\mbox{disc}(M)\geq \Omega(mn)\cdot \min\left\{p,\frac{p^{1/2}}{\sqrt{r}}\right\}.$$ We use this result to obtain a modest improvement of Lovett's best known upper bound on the log-rank conjecture. We prove that any $m\times n$ binary matrix $M$ of rank at most $r$ contains an $(m\cdot 2^{-O(\sqrt{r})})\times (n\cdot 2^{-O(\sqrt{r})})$ sized all-1 or all-0 submatrix, which implies that the deterministic communication complexity of any Boolean function of rank $r$ is at most $O(\sqrt{r})$.

翻译：给定一个$m\times n$的二元矩阵$M$，满足$|M|=p\cdot mn$（其中$|M|$表示1的个数），定义$M$的不一致性为$\mbox{disc}(M)=\displaystyle\max_{X\subset [m], Y\subset [n]}\big||M[X\times Y]|-p|X|\cdot |Y|\big|$。利用半定规划和谱技术，我们证明：若$\mbox{rank}(M)\leq r$且$p\leq 1/2$，则$$\mbox{disc}(M)\geq \Omega(mn)\cdot \min\left\{p,\frac{p^{1/2}}{\sqrt{r}}\right\}.$$ 我们利用此结果对Lovett关于对数秩猜想的最佳已知上界进行了适度改进。我们证明：任意秩至多为$r$的$m\times n$二元矩阵$M$均包含一个大小为$(m\cdot 2^{-O(\sqrt{r})})\times (n\cdot 2^{-O(\sqrt{r})})$的全1或全0子矩阵，这意味着任意秩为$r$的布尔函数的确定性通信复杂度至多为$O(\sqrt{r})$。