论文标题
有界差异{-1,1} -Matrices中的零和正方形
Zero-sum squares in bounded discrepancy {-1,1}-matrices
论文作者
论文摘要
对于$ n \ ge 5 $，我们证明，每$ n \ times n $矩阵$ m =（a_ {i，j}）$，条目中的条目为$ \ { - 1,1 \} $，并且绝对差异$ | \ mathrm {disc}（mathrm {disc}（m）对称）。 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 a split matrix is a matrix with all entries above the diagonal equal to $-1$ and all remaining entries equal至$ 1 $。特别是，我们表明，对于$ n \ ge 5 $，每个零和$ n \ times n $矩阵，条目中的条目$ \ { - 1,1 \} $包含一个零和平方。
For $n\ge 5$, we prove that every $n\times n$ matrix $M=(a_{i,j})$ with entries in $\{-1,1\}$ and absolute discrepancy $|\mathrm{disc}(M)|=|\sum a_{i,j}|\le n$ contains a zero-sum square except for the split 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 a split 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 $n\times n$ matrix with entries in $\{-1,1\}$ contains a zero-sum square.
