$(1,2)$-GDWN splits

We study impartial take away games on 2 unordered piles of finite nonnegative numbers of tokens $(x,y)$. Two players alternate in removing at least one and at most all tokens from the respective piles, according to certain rules, and the game terminates when a player in turn is unable to move. We follow the normal play convention, which means that a player who cannot move loses. In the game of Wythoff Nim, a player is allowed to remove either any number of tokens from precisely one of the piles or the same number of tokens from both. Let $\phi = \frac{1+\sqrt{5}}{2}$ and for all nonnegative integers $n$, $A_n=\lfloor\phi n \rfloor$ and $B_n=A_n+n$. The P-positions of Wythoff Nim are all pairs of piles with $A_n$ and $B_n$ tokens respectively. We study a generalization of this game called $(1,2)\G$ where, in addition to the rules of Wythoff Nim, a player has the choice to remove a positive number of tokens from one of the piles and twice that number from the other pile. We show that there is an infinite sector $\alpha \le y/x \le \alpha +\epsilon$, for given real numbers $\alpha>1$ and $\epsilon > 0$, for which each $(x,y)$ is an N-position, but that there are infinitely many P-positions for both $1\le y/x <\alpha$ and $\alpha +\epsilon < y/x$. This proves a conjecture from a recent paper. Namely, the adjoined set of moves in $(1,2)\G$ \emph{splits} the beam of slope $\phi$ P-positions of Wythoff Nim. We also provide a lower bound on the lower asymtotic density of lower pile heights of P-positions for extensions of Wythoff Nim. Suppose that $(a_i)$ and $(b_i)$, $i>0$, is a pair of so-called complementary sequences on the natural numbers which satisfy $(a_i)$ is increasing and for all $i$, $a_i<b_i$, for all $i\ne j$, $b_i-a_i\ne b_j-a_j$. Then \liminf_{n\rightarrow \infty}\frac{#\{i\mid a_i < n\}}{n} \ge \phi^{-1}.