Zero-sum squares in bounded discrepancy {-1,1}-matrices

Abstract

For n ≥ 5, we prove that every n × n {−1, 1}-matrix M = (a ij ) with discrepancy disc(M) = a ij ≤ n contains a zero-sum square except for the diagonal matrix (up to symmetries). Here, a square is a 2 × 2 sub-matrix of M with entries a i,j , a i+s,s , a i,j+s , a i+s,j+s for some s ≥ 1, and the diagonal 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 ≥ 5 every zero-sum n × n {−1, 1}matrix contains a zero-sum square.

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

“…The final lemma we will need to prove Theorem 2 is a variation on Claims 1 and 2 from [1]. The main difference between Lemma 5 and the result used by Arévalo, Montejano and Roldán-Pensado is that we will always find a square submatrix.…”

“…Throughout the rest of this paper, we will assume that all submatrices except squares are consecutive submatrices. We start by stating the following lemma from [1] which, starting from a small t ′ -diagonal submatrix M ′ , determines many entries of the matrix M. An example application is shown in Figure 1.…”

“…Lemma 3 (Claim 3 in [1]). Let M be an n × n {−1, 1}-matrix with no zerosum squares, and suppose that there is a submatrix M ′ = M[p : p+s, q : q +s] which is t ′ -diagonal for some 2 ≤ t ′ ≤ 2s − 3.…”

“…Recently, Arévalo, Montejano and Roldán-Pensado [1] initiated the study of a zero-sum variant of Erickson's problem. Here we wish to avoid zero-sum squares, squares with entries that sum to 0.…”

“…Arévalo, Montejano and Roldán-Pensado [1] proved that, except when n ≤ 4, every n × n non-diagonal {−1, 1}-matrix M with | disc(M)| ≤ n has a zero-sum square. They remark that it should be possible to extend their proof to give a bound of 2n, and they conjecture that the bound Cn should hold for any C > 0 when n is large enough relative to C. Conjecture 1 (Conjecture 3 in [1]). For every C > 0 there is a integer N such that whenever n ≥ N the following holds: every n × n non-diagonal {−1, 1}-matrix M with | disc(M)| ≤ Cn contains a zero-sum square.…”