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

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## Abstract

Let \({\mathbb {Z}}\) be the set of integers and \({\mathbb {Z}}_l\) be the set of integers modulo We will give sufficient conditions for which the equality holds. When \(k_1=k_2=\cdots =k_m=1\) and \(n=p^l\) where

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

*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 \(l=1\).## Keywords

Discrete torus No-three-in-line problem Chinese remainder theorem## Mathematics Subject Classification

05B30 11D79## Notes

### Acknowledgements

We would like to thank the anonymous referee for the comments that had helped us make several improvements to this paper.

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