TITLE: Zigzags and Collinear Lattice Points
QUESTION [5 upvotes]: A "zigzag" is an infinite planar path formed by starting at $(0,0)$ and then repeatedly moving $1$ unit up or $1$ unit right. A lattice point is a point with integral coordinates. Given a zigzag $Z$ and a positive integer $n$, must $Z$ contain (at least) $n$ collinear lattice points?

REPLY [5 votes]: In two dimensions it is fairly cheap. First observe (as it has been done already) that the bad path (if it exists) can cross any line with rational slope only finitely many times, which implies that eventually it is contained in an arbitrarily small angle $A_\varepsilon$ around some direction $\theta$ (i.e., $|y-\theta x|\le\varepsilon x$; WLOG the escape slope $\theta\le 1$, otherwise just swap $x$ and $y$). Now pick up a big $P$ and choose some $p\in\{1,\dots,P\},q\ge 0$ such that $|q-\theta p|\le 1/P$ (Dirichlet approximation theorem). Then for large $N$, the path has $\ge N/2$ points in the sector $A_{1/(pP)}\cap B(0,N)$ but for any point $(x,y)$ in this sector, we have $$|py-qx|\le p|y-\theta x|+|q-\theta p|x\le pN/(pP)+N/P\le 2N/P,$$ so there are at least $P/8$ points with the same value of $py-qx$, which are on the same line.