Zero-Sum Squares in $\{-1, 1\}$-Matrices with Low Discrepancy

Given a matrix $M = (a_{i,j})$ square is $2 \times 2$ submatrix with entries $a_{i,j}$, $a_{i, j+s}$, $a_{i+s, j}$, j +s}$ for some $s \geq 0$, and zero-sum where the sum to $0$. Recently, Arévalo, Montejano Roldán-Pensado proved that all large $n n$ $\{-1,1\}$-matrices $M$ discrepancy $|\sum a_{i,j}| \leq contain unless they are split. We improve this bound by showing at most $n^2/4$ either split or square. Since free matrices $n^2/2$ already known, asymptotically optimal.