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 ${\tau }_{m,n}\le 2gcd(m,n)$. 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 ${\left({\tau }_{z,n}\right)}_{n\in \mathbb{N}}$ 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 $(a?1)p+2$.