Skip to main content
MathOverflow

Return to Question

added 123 characters in body
Source Link
DmitryZ
DmitryZ
  • 960
  • 6
  • 17

Let $A$ be a symmetric row-column increasing $n \times n$ matrix (i.e. $A(i, j) < A(i+1,j)$ and $A(i, j) < A(i, j+1)$) with integer entries $A(i, j) \in \{1, 2, ..., n^2\}$. Moreover, let us assume that $A$ contains $\Theta(n^2)$ distinct entries, so the set of entries has positive density.

Define a "row discrepancy" for a rectangle $R=\{A(i,j); a_1 \leq i \leq a_2, b_1 \leq j \leq b_2 \}$ as $$D(R) = \frac{(a_2-a_1)(b_2-b_1)}{\max_{b_1 \leq j \leq b_2} (A(a_2, j)-A(a_1, j))}.$$ Let us also define the total "row discrepancy" $D_r$ as a supremum of $D(R)$ over all rectangles $R$. Is it true that $D_r$ must go to infinity as $n \to \infty$ for any $A$?

Let $A$ be a symmetric row-column increasing $n \times n$ matrix (i.e. $A(i, j) < A(i+1,j)$ and $A(i, j) < A(i, j+1)$) with integer entries $A(i, j) \in \{1, 2, ..., n^2\}$. Define a "row discrepancy" for a rectangle $R=\{A(i,j); a_1 \leq i \leq a_2, b_1 \leq j \leq b_2 \}$ as $$D(R) = \frac{(a_2-a_1)(b_2-b_1)}{\max_{b_1 \leq j \leq b_2} (A(a_2, j)-A(a_1, j))}.$$ Let us also define the total "row discrepancy" $D_r$ as a supremum of $D(R)$ over all rectangles $R$. Is it true that $D_r$ must go to infinity as $n \to \infty$ for any $A$?

Let $A$ be a symmetric row-column increasing $n \times n$ matrix (i.e. $A(i, j) < A(i+1,j)$ and $A(i, j) < A(i, j+1)$) with integer entries $A(i, j) \in \{1, 2, ..., n^2\}$. Moreover, let us assume that $A$ contains $\Theta(n^2)$ distinct entries, so the set of entries has positive density.

Define a "row discrepancy" for a rectangle $R=\{A(i,j); a_1 \leq i \leq a_2, b_1 \leq j \leq b_2 \}$ as $$D(R) = \frac{(a_2-a_1)(b_2-b_1)}{\max_{b_1 \leq j \leq b_2} (A(a_2, j)-A(a_1, j))}.$$ Let us also define the total "row discrepancy" $D_r$ as a supremum of $D(R)$ over all rectangles $R$. Is it true that $D_r$ must go to infinity as $n \to \infty$ for any $A$?

It's more convenient to allow entries up to $n^2$
Source Link
DmitryZ
DmitryZ
  • 960
  • 6
  • 17

Let $A$ be a symmetric row-column increasing $n \times n$ matrix (i.e. $A(i, j) < A(i+1,j)$ and $A(i, j) < A(i, j+1)$) with integer entries $A(i, j) \in \{1, 2, ..., n(n+1)/2\}$$A(i, j) \in \{1, 2, ..., n^2\}$. Define a "row discrepancy" for a rectangle $R=\{A(i,j); a_1 \leq i \leq a_2, b_1 \leq j \leq b_2 \}$ as $$D(R) = \frac{(a_2-a_1)(b_2-b_1)}{\max_{b_1 \leq j \leq b_2} (A(a_2, j)-A(a_1, j))}.$$ Let us also define the total "row discrepancy" $D_r$ as a supremum of $D(R)$ over all rectangles $R$. Is it true that $D_r$ must go to infinity as $n \to \infty$ for any $A$?

Let $A$ be a symmetric row-column increasing $n \times n$ matrix (i.e. $A(i, j) < A(i+1,j)$ and $A(i, j) < A(i, j+1)$) with integer entries $A(i, j) \in \{1, 2, ..., n(n+1)/2\}$. Define a "row discrepancy" for a rectangle $R=\{A(i,j); a_1 \leq i \leq a_2, b_1 \leq j \leq b_2 \}$ as $$D(R) = \frac{(a_2-a_1)(b_2-b_1)}{\max_{b_1 \leq j \leq b_2} (A(a_2, j)-A(a_1, j))}.$$ Let us also define the total "row discrepancy" $D_r$ as a supremum of $D(R)$ over all rectangles $R$. Is it true that $D_r$ must go to infinity as $n \to \infty$ for any $A$?

Let $A$ be a symmetric row-column increasing $n \times n$ matrix (i.e. $A(i, j) < A(i+1,j)$ and $A(i, j) < A(i, j+1)$) with integer entries $A(i, j) \in \{1, 2, ..., n^2\}$. Define a "row discrepancy" for a rectangle $R=\{A(i,j); a_1 \leq i \leq a_2, b_1 \leq j \leq b_2 \}$ as $$D(R) = \frac{(a_2-a_1)(b_2-b_1)}{\max_{b_1 \leq j \leq b_2} (A(a_2, j)-A(a_1, j))}.$$ Let us also define the total "row discrepancy" $D_r$ as a supremum of $D(R)$ over all rectangles $R$. Is it true that $D_r$ must go to infinity as $n \to \infty$ for any $A$?

Source Link
DmitryZ
DmitryZ
  • 960
  • 6
  • 17

Is a "row discrepancy" of symmetric row-column increasing matrices unbounded?

Let $A$ be a symmetric row-column increasing $n \times n$ matrix (i.e. $A(i, j) < A(i+1,j)$ and $A(i, j) < A(i, j+1)$) with integer entries $A(i, j) \in \{1, 2, ..., n(n+1)/2\}$. Define a "row discrepancy" for a rectangle $R=\{A(i,j); a_1 \leq i \leq a_2, b_1 \leq j \leq b_2 \}$ as $$D(R) = \frac{(a_2-a_1)(b_2-b_1)}{\max_{b_1 \leq j \leq b_2} (A(a_2, j)-A(a_1, j))}.$$ Let us also define the total "row discrepancy" $D_r$ as a supremum of $D(R)$ over all rectangles $R$. Is it true that $D_r$ must go to infinity as $n \to \infty$ for any $A$?