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

Abstract

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.