Zero-sum squares in $\{-1, 1\}$-matrices with low discrepancy

TL;DR

This paper improves bounds on the discrepancy of $\

Contribution

It establishes a tighter bound for the existence of zero-sum squares in low-discrepancy $\\{-1,1\ ext{"} matrices, extending previous results and approaching optimality.

Findings

Matrices with discrepancy at most $n^2/4$ contain zero-sum squares or are split.

The bound $n^2/4$ is asymptotically optimal for zero-sum square existence.

Large matrices with discrepancy up to $n^2/2$ can be zero-sum square free.

Abstract

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 $s \geq 1$, and a zero-sum square is a square where the entries sum to $0$. Recently, Ar\'evalo, Montejano and Rold\'an-Pensado 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.