On No-Three-In-Line Problem on m-Dimensional Torus

Abstract

Let ${\mathbb {Z}}$ be the set of integers and ${\mathbb {Z}}_l$ be the set of integers modulo l. A set $L\subseteq T={\mathbb {Z}}_{l_1}\times {\mathbb {Z}}_{l_2}\times \cdots \times Z_{l_m}$ is called a line if there exist ${\mathbf {a}},{\mathbf {b}}\in T$ such that $L=\{ {\mathbf {a}}+t{\mathbf {b}}\in T\ :\ t\in {\mathbb {Z}} \}$. A set $X\subseteq T$ is called a no-three-in-line set if $\vert X\cap L\vert \le 2$ for all the lines L in T. The maximum size of a no-three-in-line set is denoted by $\tau \left( T \right)$. Let $m\ge 2$ and $k_1,k_2,\ldots ,k_m$ be positive integers such that $\gcd (k_i,k_j)=1$ for all i, j with $i\ne j$. In this paper, we will show that

$$\begin{aligned} \tau \left( {\mathbb {Z}}_{k_1n}\times {\mathbb {Z}}_{k_2n}\times \cdots \times Z_{k_mn} \right) \le 2n^{m-1}. \end{aligned}$$

We will give sufficient conditions for which the equality holds. When $k_1=k_2=\cdots =k_m=1$ and $n=p^l$ where p is a prime and $l\ge 1$ is an integer, we will show that equality holds if and only if $p=2$ and $l=1$, i.e., $\tau \left( {\mathbb {Z}}_{p^l}\times {\mathbb {Z}}_{p^l}\times \cdots \times Z_{p^l} \right) =2p^{l(m-1)}$ if and only if $p=2$ and $l=1$.