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

Abstract: For $n\ge 5$, we prove that every $n\times n$ $\{-1,1\}$-matrix $M=(a_{ij})$ with discrepancy $\text{disc}(M)=\sum a_{ij} \le n$ contains a zero-sum square except for the diagonal matrix (up to symmetries). Here, a square is a $2\times 2$ sub-matrix of $M$ with entries $a_{i,j}, a_{i+s,s}, a_{i,j+s}, a_{i+s,j+s}$ for some $s\ge 1$, 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 $n\ge 5$ every zero-sum $n\times n$ $\{-1,1\}$-matrix contains a zero-sum square.

Joint work with Edgardo Roldán-Pensado and Alma Arévalo.

number theory

Audience: researchers in the topic

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Combinatorial and additive number theory (CANT 2021)

Series comments: This is the nineteenth in a series of annual workshops sponsored by the New York Number Theory Seminar on problems in combinatorial and additive number theory and related parts of mathematics.

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