Geometric Variants of the Gale--Berlekamp Switching Game

The Gale-Berlekamp switching game is played on the following device: $G_n=\{1,2,\ldots,n\} \times \{1,2,\ldots,n\}$ is an $n \times n$ array of lights is controlled by $2n$ switches, one for each row or column. Given an (arbitrary) initial configuration of the board, the objective is to have as many lights on as possible. Denoting the maximum difference (discrepancy) between the number of lights that are on minus the number of lights that are off by $F(n)$, it is known (Brown and Spencer, 1971) that $F(n)= \Theta(n^{3/2})$, and more precisely, that $F(n) \geq \left( 1+ o(1) \right) \sqrt{\frac{2}{\pi}} n^{3/2} \approx 0.797 \ldots n^{3/2}$. Here we extend the game to other playing boards. For example: (i)~For any constant $c>1$, if $c n$ switches are conveniently chosen, then the maximum discrepancy for the square board is $\Omega(n^{3/2})$. From the other direction, suppose we fix any set of $a$ column switches, $b$ row switches, where $a \geq b$ and $a+b=n$. Then the maximum discrepancy is at most $-b (n-b)$. (ii) A board $H \subset \{1,\ldots,n\}^2$, with area $A=|H|$, is \emph{dense} if $A \geq c (u+v)^2$, for some constant $c>0$, where $u= |\{x \colon (x,y) \in H\}|$ and $v=|\{y \colon (x,y) \in H\}|$. For a dense board of area $A$, we show that the maximum discrepancy is $\Theta(A^{3/4})$. This result is a generalization of the Brown and Spencer result for the original game. (iii) If $H$ consists of the elements of $G_n$ below the hyperbola $xy=n$, then its maximum discrepancy is $\Omega(n)$ and $O(n (\log n)^{1/2})$.