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

Tom Johnston¹

¹University of Bristol and Heilbronn Institute for Mathematical Research

The electronic journal of combinatorics, Tome 30 (2023) no. 2

Résumé

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 $s \geq 0$, and a zero-sum square is a square where the entries sum to $0$. Recently, Arévalo, Montejano and Roldán-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.

Détail

DOI : 10.37236/10928

Classification : 05D10
Mots-clés : Erickson's problem, zero-sum problems