Proof of a conjecture of Johnston on binary matrices with low discrepancy

A square in a matrix M = (a i, j ) is a 2 × 2 submatrix S with row and column indices {i, i + s} and { j, j + s}, respectively. A square is zero-sum if the sum of its entries equals 0. In 2021, Arévalo, Montejano and Roldán-Pensado proved that all large binary matrices M over {1, -1} of order n with absolute discrepancy | ∑ a i, j | ≤ n contain a zero-sum square, unless they are split. In 2023, Johnston showed that the same conclusion follows from the much weaker hypothesis | ∑ a i, j | ≤ n 2 /4. He further conjectured that the same conclusion would still follow from the weakest possible hypothesis, namely -2 if n is odd. We prove this conjecture here. We also address a related problem raised by Johnston in the rectangular case.