On No-Three-In-Line Problem on <italic>m</italic>-Dimensional Torus.

Let Z<inline-graphic></inline-graphic> be the set of integers and Zl<inline-graphic></inline-graphic> be the set of integers modulo <italic>l</italic>. A set L⊆T=Zl1×Zl2×⋯×Zlm<inline-graphic></inline-graphic> is called a line if there exist a,b∈T<inline-graphic></inline-graphic> such that L={a+tb∈T:t∈Z}<inline-graphic></inline-graphic>. A set X⊆T<inline-graphic></inline-graphic> is called a no-three-in-line set if |X∩L|≤2<inline-graphic></inline-graphic> for all the lines <italic>L</italic> in <italic>T</italic>. The maximum size of a no-three-in-line set is denoted by τT<inline-graphic></inline-graphic>. Let m≥2<inline-graphic></inline-graphic> and k1,k2,…,km<inline-graphic></inline-graphic> be positive integers such that gcd(ki,kj)=1<inline-graphic></inline-graphic> for all <italic>i</italic>, <italic>j</italic> with i≠j<inline-graphic></inline-graphic>. In this paper, we will show that τZk1n×Zk2n×⋯×Zkmn≤2nm-1.<graphic></graphic>We will give sufficient conditions for which the equality holds. When k1=k2=⋯=km=1<inline-graphic></inline-graphic> and n=pl<inline-graphic></inline-graphic> where <italic>p</italic> is a prime and l≥1<inline-graphic></inline-graphic> is an integer, we will show that equality holds if and only if p=2<inline-graphic></inline-graphic> and l=1<inline-graphic></inline-graphic>, i.e., τZpl×Zpl×⋯×Zpl=2pl(m-1)<inline-graphic></inline-graphic> if and only if p=2<inline-graphic></inline-graphic> and l=1<inline-graphic></inline-graphic>. [ABSTRACT FROM AUTHOR]