论文标题
零和正方形$ \ { - 1，1，1 \} $ - 矩阵差异低
Zero-sum squares in $\{-1, 1\}$-matrices with low discrepancy
论文作者
论文摘要
给定一个矩阵$ m =（a_ {i，j}）$ a square是一个$ 2 \ times 2 $ subbatrix，带有条目$ a_ {i，j} $，$ a_ {i，j+s} $，$ a__ {i+s，j} $，$ a_+s，$ a_+s，j+s，a z a z a z a z $ s a z $总和$ 0 $。最近，Arévalo，Montejano和Roldán-Pensado证明了所有大型$ n \ times n $ $ \ { - 1,1 \} $ - 矩阵$ m $，带有差异\ leq n $除非分开，否则包含零和正方形。我们通过表明所有大型$ n \ times n $ $ \ { - 1,1 \} $ - 矩阵$ m $具有差异最多$ n^2/4 $是分裂或包含零和正方形的零件。由于最多$ n^2/2 $的零和平方矩阵已经知道，因此该界限在渐近上是最佳的。
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, 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.
