No-three-in-line problem on a torus: periodicity

Let $\tau_{m,n}$ denote the maximal number of points on the discrete torus (discrete toric grid) of sizes $m \times n$ with no three collinear points. The value $\tau_{m,n}$ is known for the case where $\gcd(m,n)$ is prime. It is also known that $\tau_{m,n} \leq 2\gcd(m,n)$. In this paper we generalize some of the known tools for determining $\tau_{m,n}$ and also show some new. Using these tools we prove that the sequence $(\tau_{z,n})_{n \in \mathbb{N}}$ is periodic for all fixed $z > 1$. In general, we do not know the period; however, if $z = p^a$ for $p$ prime, then we can bound it. We prove that $\tau_{p^a,p^{(a-1)p+2}} = 2p^a$ which implies that the period for the sequence is $p^b$ where $b$ is at most $(a-1)p+2$.