Zero-sum squares in \(\{-1, 1\}\)-matrices with low discrepancy (Q6046229)

Summary: Given a matrix \(M = (a_{i,j})\) a square is a \(2 \times 2\) submatrix with entries \(a_{i,j}, a_{i, j+s}, a_{i+s, j}, a_{i+s, j +s}\) for some \(s \geqslant 0\), and a zero-sum square is a square where the entries sum to \(0\). Recently, \textit{A. R. Arévalo} et al. [ibid. 28, No. 4, Research Paper P4.15, 14 p. (2021; Zbl 1476.05193)] proved that all large \(n \times n \{-1,1\}\)-matrices \(M\) with discrepancy \(|\sum a_{i,j}| \leq n\) contain a zero-sum square unless they are split. We improve this bound by showing that all large \(n \times n \{-1,1\}\)-matrices \(M\) with discrepancy at most \(n^2/4\) are either split or contain a zero-sum square. Since zero-sum square free matrices with discrepancy at most \(n^2/2\) are already known, this bound is asymptotically optimal.