For n ≥ 5, we prove that every n × n {−1, 1}-matrix M = (a ij ) with discrepancy disc(M) = a ij ≤ n contains a zero-sum square except for the diagonal matrix (up to symmetries). Here, a square is a 2 × 2 sub-matrix of M with entries a i,j , a i+s,s , a i,j+s , a i+s,j+s for some s ≥ 1, and the diagonal matrix is a matrix with all entries above the diagonal equal to −1 and all remaining entries equal to 1. In particular, we show that for n ≥ 5 every zero-sum n × n {−1, 1}matrix contains a zero-sum square.
For $n\geqslant 5$, we prove that every $n\times n$ matrix $\mathcal{M}=(a_{i,j})$ with entries in $\{-1,1\}$ and absolute discrepancy $\lvert\mathrm{disc}(\mathcal{M})\rvert=\lvert\sum a_{i,j}\rvert\leqslant n$ contains a zero-sum square except for the split matrices (up to symmetries). Here, a square is a $2\times 2$ sub-matrix of $\mathcal{M}$ with entries $a_{i,j}, a_{i+s,s}, a_{i,j+s}, a_{i+s,j+s}$ for some $s\geqslant 1$, and a split matrix is a matrix with all entries above the diagonal equal to $-1$ and all remaining entries equal to $1$. In particular, we show that for $n\geqslant 5$ every zero-sum $n\times n$ matrix with entries in $\{-1,1\}$ contains a zero-sum square.
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